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

UNIVERSITY OF CALIFORNIA L o s Angeles O p t i m a l I n v e s t m e n t a n d Consumption S t r a t e g i e s f o r a C l a s s of Utility F u n c t i o n s A d i s s e r t a t i o n s u b m i t t e d i n p a r t i a l s a t i s f a c t i o n of the r e q u i r e m e n t s f o r t h e d e g r e e Doctor of P h i l o s o p h y i n B u s i n e s s Administration Nils H e m m i n g H a k a n s s o n Committee in charge: P r o f e s s o r G e o r g e W Brown, C h a i r m a n P r o f e s s o r Leo B r e i m a n Professor Jack Hirshleifer Professor Jacob Marschak P r o f e s s o r J , F r e d Weston Copyright by N I L S HEMMING HAKANSSON T e diaclerta;tion of N i b Hemming Hakaneson is h appsoved, and it is acceptable In quality and form for publication on mlcrof'ilm: * r Committee Chairman Unlveraity of California, Ins Augeles TABLE O F CONTENTS LIST O F FIGURES ACKNOWLEDGMENTS VITA ABSTRACT v LIST O F TABLES vi vii viii ix Page Chapter Investment vs Consumption The Utility F u n c t i o n THE BUILDING BLOCKS 1 2 1 2 3 Koopmans' Impatience Study , P r o p e r t i e s and Limitations of the Utility Function Note on the Boundedness of the Utility Function Opportunities f o r Decision The Opportunity Set 1 3 3 Opportunities f o r Non-Capital Income Productive Investment Opportunities F i n a n c i a l Opportunities THE MODEL AND ITS IMPLICATIONS 2 2, Derivation of the ~ a s i Model c The Solution When u(xy) = u ( x ) / u ( ~ ) 2.3.1 Model I 2.3.2 ModelII The Solution When u ( x y ) = u(x) + u ( y ) The Solution Wher, u ( x t y ) = u ( x ) ( u ( ~ , , 1) S u m m a r y o f Notation P r o p e r t i e s of the Optimal Consumption Strategies 6 2 Effect of Impatience R a t e Effect of Risk A v e r s i o n Index Effect of the " F a v o r a b l e n e s s " of the Investment Opportunities The Behaviour of Capital iii T A B L E O F CONTENTS ( C o n t ) 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 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 Rate 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 and Lending 2.10 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 Strategies Generalizations 10 F i n i t e H o r i z o n 10 Non-Constant Non- C a p i t a l I n c o m e 10 Stream T i m e -Dependent P r o b a b i l i t y Distributions 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 11 11 2 1 Firms B a s e s f c r t h e F o r m a t i o n of The Firm's Objective and I t s O p t i m a l Capital Structure The 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 1 APPLICATIONS AND E X A M P L E S Individual De c i s ion-Making 3 The Bal.aaced Mutual F u n d IV Endowed E d u c a t i o n a l and C h a r i t a b l e O r g a n i z a t i o n s F i s h e r ' s Model of t h e Individual Consumptio:? Models C l a s s i c a l Models P h e l p s ! Mode1 RELATION T O OTHER MODELS 4 I n v e s t m e n t Models The Me a r - V a r i a n c e A p p r o a c h C h a n c e - C o s s t r a i n e d Models Long-Ru-2 ~ n v e s t m e n tModels O t h e r I n v e s t m e n t Models 4 The S t a t e - P r e f e r e z c e A p p r o a c h : A Brief Comment Summary BIBLIOGRAPHY LIST O F T A B L E S PAGE TABLE I 11 Z q u i v a l e n t Consumption P r o g r a m s When u ( c ) = log c * O p t i m a l Allocation of C a p i t a l (A; E a c h D e c i s i o n P o i n t ) f o r S e l e c t e d C a p i t a l P o c i l i o n s When u ( c ) = a = 88, y = $10, 000 i n E a c h P e r i o d , a n d r , P , P3' P4 a r e a s i n ( - ) fi 111 O p t i m a l Allocation of; C a p i t a l (At E a c h D e c i s i o n P o i n t ) f o r S e l e c t e d C a p i t a l P o s i t i o n s When ~ J ( C ) = - c - , cu = 88, y = $10, 000 i n E a c h P e r i o d , a n d r , P2, P3, P4, a r e as i n ( - ) IV V O p t i m a l A l l o c a t i o n 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 n t ) f o r S e l e c t e d C a p i t a l P o s i t i o n s When u ( c ) = log c , a, = 88, y = $10, 000 i n E a c h P e r i o d , a n d r , P2, P3, P4, a r e a s i n ( - ) ' O p t i m a l Allocation of C a p i t a l ( A t E a c h D e c i s i o n P o i n t ) f o r S e l e c t e d C a p i t a l P o s i t i o n s When u ( c ) = - e - , 0 c , Q = 99$ y = $ l o 000 i n Ea.ch P e r i o d , r = , and P m a y A s s u m e E a c h of t h e Values 96 and 1 w ~ t l ? r o b a b i l i t y P a VI E f f e c t of I m p a t i e n c e R a t e o n O p t i m a l Allocation af C a p i t a l ( A t E a c h D e c i s i o n P o i n t ) When u ( c ) = - c - , y = $10, 000 i n E a c h P e r i o d , a n d r , P , P , P4, a r e a s in (3-1) VII E f f e c t of Size of (Con.stant) N o n - c a p i t a l I n c o m e S t r e a m o n O p t i m a l A ? l o c a t i o ~ C a p i t a l (At E a c h of D e c i s i o n P o i s t ) When U(C) = log c , a = 90, a n d r , P2, B3, P4, a r e a s i n ( - ) VIII Normative Investment and Consumption Models: A Comparative Summary LIST O F FIGURES PAGE FIGURE Zones of Impatience F i s h e r ' s Solution 15 104 ACKNOWLEDGMENTS While the r e s e a r c h r e p o r t e d i n this 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 These 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 he h a s s e t have not only benefitted this study but have profoundly influenced m y own development Highly generous i n the 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 been a continual s o u r c e of inspiration I a m a l s o indebted to P r o f e s s o r J a c o b M a r s c h a k f o r m a n y e a r l y suggestions and valuable c o m m e n t s , and to P r o f e s s o r s Leo B r e i m a n , J a c k H i r s h l e i f e r , J a m e s Jackson, and J F r e d Weston f o r c o n s t r u c tive c r i t i c i s m s I a m g r a t e f u l to the F o r d Foundation f o r ca.rrying the fi.nancia1 burden 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 Fellowships and one D i s s e r t a t i o n Fellowship, and to the 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 providing the n e c e s s a r y computational facj-lities Finally, I wish to p e x p r e s s m y s i n c e r e appreciation to the Western Management S,' i e m e Institute of UCLA and the RAND Corporation f o r typing portions of the 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 Harrington for t h e i r excellent typi.ng of the final m a n u s c r i p t June 2, 1937 1956 - B o r n - M a r b y , Sweden S t u d e n t e x a m e n , s t e r S u n d s ~ g r o v e r k ,~ s t e r s u n d ,Sweden 1956 1958 - Royal Swedish C o r p s of E n g i n e e r s - B S , U n i v e r s i t y of O r e g o n 1958-1960 - IBM R e s e a r c h A s s i s t a n t , W e s t e r n D a t a P r o c e s s i n g C e n t e r , U n i v e r s i t y of C a l i f o r n i a , Los Angeles 1960 - M, B A , U n i v e r s i t y of C a l i f o r n i a , L o s A n g e l e s 1960-1963 - Staff a c c o u n t a n t (1960-1962), C o n s u l t a n t (1962-1963), A r t h u r Young a n d Company, C e r t i f i e d P u b l i c A c c o u n t a n t s , Los A n g e l e s , C a l i f o r n i a 1962 - C e r t i f i e d P u b l i c Accountant ( C a l i f o r n i a ) 1963-1966 - F o r d F e l l o w , G r a d u a t e School of B u s i n e s s A d m i n i s t r a t i o n , U n i v e r s i t y of C a l i f o r n i a , Los A n g e l e s 1964 - P o s t g r a d u a t e R e s e a r c h A s s i s t a n t XI., W e s t e r n M a n a g e m e n t S c i e n c e I n s t i t u t e , U n i v e r s i t y of C a l i f o r n i a , L o s A n g e l e s 1965 - C o n s u l t a n t , t h e RAND C o r p o r a t i o n , S a n t a Monica California F I E L D S O F STUDY Major Field: Business Administration Studies i n M a t h e m a t i c a l Methods P r o f e s s o r s G e o r g e W * Brown, R i c h a r d V E v a n s , a n d J a m e s R J a c k s o n Studies i n Economics P r o f e s s o r s J a c o b M a r s c h a k a n d F r a n k E Norton, P r o f e s s o r K a r l B o r c h , N o r w e g i a n School of Economi.cs! B e r g e n S t u d i e s i n Accounting P r o f e s s o r s A B C a r s o n , P a u l K i r c h e r , a n d H a r r y Simdns Studies in Management Theory P r o f e s s d r H a r o l d Koontz vi'ii ABSTRACT O F THE DISSERTATION Optimal Investment 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 son Doctor of Philosophy i n B u s i n e s s Administration University of California, Los Angeles, 1966 P r o f e s s o r George W Brown, C h a i r m a n This r e s e a r c h f o r m a l i z e s Irving F i s h e r ' s m o d e l of the individuai under r i s k , and r e p r e s e n t s a t the s a m e t i m e a g e n e r a l i z a t i o n of P h e i p s ' m o d e l of p e r s o n a l saving (-c o n o m e t r i c a , October 1962) The E-objective of the individual i s postulated to be the maximization of expected utility f r o m consumptior, o v e r t i m e where the horizon i s a r b i t r a r i l y distant The individual's r e s o u r c e s c o n s i s t of a n i n i t i a l c a p i - tal p o s i t i o ~ (which m a y be ~ e g a t i v e and a noc-capital income 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 p o s s e s s any t i m e - s h a p e The individual f a c e s both financial opportunities (borrowing and lending) and a n a r b i t r a r y number of productive inve s t m e a t opportu.nlties choice, the sequential n a t u r e of the investment p r o c e s s i s recognized explicitly instead of not a t a l l , the decision concerning how much to invest a n d / o r to borrow i s endogenous r a t h e r than exogenous, and the individu&I1sn.on-capital income s t r e a m i s a n i n t e g r a l p a r t of the model i n s t e a d of being outside it F u r t h e r m o r e , Models I-IVp o s s e s s none of the drawbacks of the m e a n - v a r i a n c e models such a s i n c r e a s i n g r i s k a v e r s i o n and d e c r e a s i n g uti.1i.t~ capital beyond a c e r t a i n capital level of Chance-Constrained Models Probably the m o s t well-knowr, investment model based on the method of chance c o n s t r a i n t s i s that of Naslund and Whinston They consider the c a s e of a n individual with a known non-capital income s t r e a m who h a s - a d ~ a r- decided how much to spend on consumption up e to a specified decision point n The objective i s postulated i o be the maximizati.on of the s u m of the expected gai.ns f r o m investments i n e a c h p e r i o d up to the horizon (deci.sion point n ) subject to two constraints The f i r s t c o n s t r a i n t p l a c e s a n upper bound on the probability that a capital 10s s i n a given p e r i o d m a y exceed a p r e s c r i b e d l i m i t The second c o n s t r a i n t r e q u i r e s invested capital i n ea.ch p e r i o d to r e main, with a given (mini.mum)probabi.l.ity, below a 1.j.mj.t d e t e r m i n e d b y the capital gains accumulated s o f a r and the accumulated net s a v - ings resulting f r o m the nor,-capital income a f t e r consumption r e q u i r e - , m e n t s have been satisfied In o u r notation the p r o b l e m m a y be s t a t e d as B e r t i l Naslund and Andrew Whinston, "A Model of Multi.- , P e r i o d Investment Under Uncertainty," Management Science, January 1962 IL Max E l s u b j e c t to Since b o r r o w i n g a n d lending a r e not c o n s i d e r e d by Naslund a n d Whinston, the f i n a n c i a l o p p o r t u n i t i e s ( i = ) h a v e b e e n l e f t out i n t h e representation above A m o d e l s i m i l a r to the one d e s c r i b e d h a s a l s o b e e n developed by H i l l i e r T h e b a s i c d i f f e r e n c e between the c h a n c e - c o n s t r a i n e d m o d e l s a n d Models I - I V i s t h a t the f o r m e r l e a v e the d e t e r m i n a t i o n of how m u c h s h o u l d be c o n s u m e d a n d what r i s k s to a c c e p t o u t s i d e the f o r m a l m o d e l T h u s , t r a d e o f f s i t h e s e v a r i a b l e s c a n only be e v a l u a t e d i n f o r m a l l y z e v e n though t h e s e c o ~ s i d e r a t i o r sa r e no m o r e s u b j e c t i v e t h a n f o r ex-ample model t h e p r o b a b i l i t y d i s t r i b u t i o n s of r e t u r n which - a r e p a r t of t h e In Models I - i V , o n t h e o t h e r ha2d, t h e s e t r a d e o f f s a r e a u t o - m a t i c a l l y evalua-ted s i c c e a u.tility f u z c t i o n (of c o n s u m p t i o n j i s p r e s e n t which a p p r a i s e s a l l p o s s i b l e ( u l t i m a t e ) o u t c o m e s of a l l p o s s i b l e d e c i sions - Thus the r e l a t i o n between t h e c h a n c e c o n s t r a i n e d i n v e s t m e n t F r e d e r i c k H i l l i e r , " The E v a l u a t i o n of Risky I n t e r r e l a t e d Lnves t m e n t s , " T e c h ~ i c a R e p o r t No 73, Stanford, C a l i f o r n i a , Departmen-t of l S t a t i s t i c s , Stanford U n i v e r s i t y , Suly 1964, p p 45-55 models of Naslund and Whinston and of KiTli.er and Models 1-IV i s analogous to that between c l a s s i c a l and m o d e r n d e c i s i o n theory 3 Long-Run I n v e s t m e n t Models L o n g - r u n investment models e s s e n t i a l l y f a l l into two c l a s s e s : t h o s e which s t r i v e to m a k e t h e l o n g - r u n c a p i t a l position a s f a v o r a b l e a s pos s i b l e and l-i~oze which s t r i v e to m a x i m i z e the probability of surviving infinitely Models which m a x i m i z e the long-run c a p i t a l position Consider )I the c a s e i n which i t i s d e s i r e d to m a x i m i z e E [ G ( ~ x l ] a t s o m e f u t u r e J d e c i s i o n point j when x ji-1 (pi - r ) z i- r x = i =2 =J j' j = 2, , and > 0, that i s , all r e t u r n s a r e m a d e available f o r r e i n v e s t m e n t i n t h e next p e r i o d I t h a s t h e n been shown that when i i ( x ) h a s the f o r m J (2-15) ( - ) , o r ( - ) , 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 a r e of the f o r m z (x)= 13 p.((P ), r ) x f o r a l l j w h e r e pi i s a proportion In o t h e r A w o r d s , 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 y c a l l s f o r investing a p r o p o r tion of c u r r e n t c a p i t a l ; : n e a c h opportunity, the p r o p o r t i o n s being dependent only upon the dj.str bution function of r e t u r n s As m a y be s e e n f r o m (2-351, (2-191, and (2-431, this p r o p e r t y of the optim a l i n v e s t m e n t s t r a t e g i e s i s not l o s t wher, the objective i s t h e m a x i m i z a t i o n of utility f r o m I F o r a c o m p a r i s o n between c l a s s i c a l s t a t i s t i c s and m o d e r n d e c i s i o n t h e o r y , s e e , f o r exa.mple, Du2can Luce and Howard Raiffa, - cit., op pp 318-324 Thomas F e r g u s o n "Betting S y s t e m s Which Minimize t h e Probability l of Ruin, " J o u r n a l of t h e Society of I ~ d u s t r i a and Applied M a t h e m a t i c s , S e p t e m b e r 1965, p 799 c o n s u m p t i o n o v e r t i m e ( g i v e n t h a t t h e o n e - p e r i o d utility function of c o n s u m p t i o n is of t h e f o r m ( - 5), ( - ) , o r ( - ) ) a n d t h e individual r e c e i v e s a non-capital income s t r e a m t h e utility function G(x) = -e - m( y > Analogously, w e find t h a t when O ) , t h e p r o p e r t y of nondependence o n x which c h a r a c t e r i z e s the o p t i m a l s t r a t e g y i n the l o n g - r u n i n v e s t ment model a l s o holds i n Model IV T h e p a r t i c u l a r m o d e l c a l l i n g f o r t h e m a x i m i z a t i o n of h a s b e e n c o n s i d e r e d by s e v e r a l a u t h o r s ,including K e l l e y , Bellman, and ~ a t a n g l lo^ x./xl ] I Br eiman, The o p t i m a l s t r a t e g y i n t h i s c a s e c a l l s f o r m a x i m i z i n g t h e e x p e c t e d l o g a r i t h m of ca.pita1 ( a t t h e end of t h e p e r i o d ) in each period B r e i m a n a l s o found t h a t t h i s policy is a s y m p t o t i c a l l y o p t i m a l when t h e o b j e c t i v e i s to m i n i m i z e t h e e x p e c t e d t i m e t o r e a c h a f i x e d l e v e l of r e s o u r c e s M o r e o v e r , t h i s s t r a t e g y t u r ~ o u t to b e s a s y m p t o t i c a l l y o p t i m a l when the o b j e c t i v e is to m a x i m i z e t h e e x p e c t e d g r o w t h of c a p i t a l As w a s noted i n , a n i n d i v i d u a l who o b e y s Model 111 ( w h e r e u ( c ) = log c ) w i l l a l s o b e h a v e s o a s to m a x i m i z e t h e e x p e c t e d g r o w t h r a t e of t h e c a p i t a l ( r e m a i n i n g a f t e r the a l l o t m e n t to c u r r e n t I b i d , p 800 J L K e l l e y , " A New I n t e r p r e t a t i o n of I n f o r m a t i o n R a t e , B e l l S y s t e m T e c h n i c a l J o u r n a l , 35, 1956 - L e o B r e i m a n , " O p t i m a l Ga.mbling S y s t e m s f o r F a v o r a b l e G a m e s , " F o u r t h B e r k e l e y s y - s i u m o n I%obabilityand M a t h e m a t i c- a t i s mpo a l St t i c s , ~ e r k e l e ~ ,n i v e r s i t y of C a l i f o r n i a P r e s s , 1961 U -4 R i c h a r d B e l l m a n , Adaptive C o n t r o l P r o c e s s e-, P r i n c e t o n , P r i n c e t o n s University P r e s s , , pp 222-229 H e n r y ~ a t a n g ," C r i t e r i a f o r Choice Among Risky Vyntur e s , I - u r n a l Jo of P o l i t i c a l Economv A ~ r i 9 , and H e n r v L a t a n e " I n v e s t m e n t l C r i t e r i a : A T h r e e z i s e t ' P o r t f o l i o B a l a n c e ~ o d e l " of E c o , keview n o m i c s a n d S t a t i s t i c s , N o v ~ r n h e r1963 B r e i m a n , " O p t i m a l Gambling S y s t e m s See B r e i m a n , " O p t i m a l Gambling S y s t e m s "; B r e i m a n , " I n v e s t m e n t P o l i c i e s , "; a n d Brown, op c i t , c o n s u m p t i o n ) plus t h e c a p i t a l i z e d value of h i s ( c e r t a i n ) n o n - c a p i t a l income stream Survival models F'erguson1 a n d ruel love' h a v e examined the p r o b l e m of o p t i m a l i n v e s t m e n t b e h a v i o u r when t h e r e is a f i x e d c o s t - o f living c h a r g e (con.sumption l e v e l ) but no b o r r o w i n g a n d no n o n - c a p i t a l income stream They p o s t u l a t e the o b j e c t i v e of the individual to be t h e m a x i m i z a t i o n of t h e p r o b a b i l i t y of s u r v i v i n g infinitely, t h a t i s , of being a l w a y s a b l e to p a y the c o s t - o f - l i v i n g c h a r g e I n t h e s e m o d e l s , then, t h e c o n s u m p t i o n l e v e l i s f i x e d a n d e x o g e n e ous H o w e v e r , t h e r e i s c l e a r l y a tradeoff p o s s i b i l i t y b e t w e e n t h e l e v e l of c o n s u m p t i o n a n d t h e s u r v i v a l p r o b a b i l i t y s i n c e the l a t t e r i s a f u n c t i o n of t h e f o r m e r a n d i t c l e a r l y m a k e s a d i f f e r e n c e -you l i v e , i e , how how m u c h you c o n s u m e while you s u r v i v e By i n t r o d u c i n g a utility f u n c t i o n defined f o r all c o n s u m p t i o n l e v e l s a n d m a k i n g t h e l e v e l of c o n s u m p t i o n a d e c i s i o n v a r i a b l e , Models I-IV, while not c o n c e r n e d with s u r v i v a l e x p l i c i t l y , n e v e r t h e l e s s h a v e t h e notion of s u r v i v a l built i n T h e notion of s u r v i v a l i s i n f a c t i m p l i c i t i n the u t i l i t y function (of c o n sumption) itself If w e a s s o c i a t e s u r v i v a l w i t h p o s i t i v e c o n s u m p t i o n i n e a c h p e r i o d , i t w a s s h o w n i n t h a t u c l e s s t h e i n d i v i d u a l s t a r t s o u t i n t h e t r a p p i n g s t a t e ( i n which c a s e h e would p e r i s h i m m e d i a t e l y ) , he w i l l s u r v i v e infinitely with p r o b a b i l i t y i n Models I1 a n d 131 while t h e s u r v i v a l p r o b a b i l i t y m a y be l e s s t h a n i n Model I Since t h e s e A J T r u e l o v e , "A M u l t i s t a g e S t o c h a s t i c I n v e s t m e n t P r o c e s s , " -RM4 - P R , Santa Monica, T h e RAND C o r p o r a t i o n , M a r c h 1964 i m p l i c a t i o n s a r e d i r e c t l y r e l a t e d to t h e l o w e r bound of the o n e - p e r i o d utility function of consumption ( s e e C o r o l l a r y l ) , we find that individuals whose utility functions have no lower bound i n f a c t p l a c e a p r e m i u m on s u r v i v a l 4 Other Investment Models All m o d e l s d i s c u s s e d s o f a r s h a r e the c h a r a c t e r i s t i c t h a t i n v e s t m e n t d e c i s i o n s a r e m a d e a t specified, d i s c r e t e points i n t i m e How- e v e r , m o d e l s have a l s o been c o n s t r u c t e d i n which i n v e s t m e n t o p p o r tunities a r r i v e r a n d o m l y i n t i m e Unless i m m e d i a t e l y a c c e p t e d , e a c h such o p p o r t u ~ l t ys considered l o s t f o r e v e r i The p r o b l e m then be- c o m e s to find o p t i m a l d e c i s i o n r u l e s f o r accepting a n d r e j e c t i n g oppor- t u n i t i e s , which a r e g e n e r a l l y viewed a s l o n g - t e r m i n n a t u r e , s o a s to have funds a v a i l a b l e f o r highly f a v o r a b l e o p p ~ r t u n i t i s which have not e yet a p p e a r e d while a t t h e s a m e t i m e taking advantage of a s m a n y opportunities a s possible and by Kaufman This p r o b l e m h a s b e e n e x a m i n e d by F i s h e r W h i l e consumption and borrowing a n d lending a r e not c o n s i d e r e d , F i s h e r does c o n s i d e r t h e c a s e when the i n v e s t o r r e c e i v e s a non-.capital i n c o m e s t r e a m Since no meaningful b a s i s f o r c o m p a r i n g t h e s e m o d e l s with Models - I V s e e m s to e x i s t , we s h a l l not review t h e s e m o d e l s f u r t h e r How- e v e r , It a p p e a r s that Models - i V m a y w e l l g e n e r a l i z e to the r a n d o m a r r i v a l situation f o r the c a s e i which the i n v e s t m e n t opportunities a r e n J a m e s F i s h e r , "A C l a s s of Stochastic Investment P r o b l e m s , " Operations R e s e a r c h , J a n u a r y - F e b r u a r y 1961, - Gordon Kaufman, "Sequential I n v e s t m e n t Analysis Under Uncer tainty, " J o u r n a l of B u s i n e s s , J a n u a r y 1963 J a m e s F i s h e r , z - _c_ t p i governed by a Markov p r o c e s s i n which the p r e v a i l i n g s t a t e i s not n e c e s s a r i l y known a t the t i m e of decision THE S T A T E - P R E F E R E N C E APPROACH: A B R I E F COMMENT This study would not be c o m p l e t e without a t l e a s t a b r i e f m e n t i o n of the s t a t e - p r e f e r e n c e a p p r o a c h to decision-making under u n c e r t a i n t y This a p p r o a c h t a k e s cognition of t h e f a c t that p r e f e r e n c e s m a y depend upon which of s e v e r a l p o s s i b l e s t a t e s of the w o r l d obtains a t a given t i m e i n the future It t h e r e f o r e r e p r e s e n t s a n i m p o r t a n t s t e p t o w a r d a m o r e r e a l i s t i c t h e o r y of i n t e r t e m p o r a l d e c i s i o n s i n c e the utility function would now b e t i m e - s t a t e -dependent r a t h e r than just t i m e -dependent a s i n t h e c a s e of Models I-IV, f o r example The pioneering w o r k i n this a r e a i s that of A r r o w ; i m p o r t a n t co.ntribution.s have a l s o been m a d e by H i r s h l e i f e r 9 No d i r e c t application of t h i s a p p r o a c h h a s been m a d e to t h e p r o b l e m a d d r e s s e d i n this study SUMMARY The p r i n c i p a l c h a r a c t e r i s t i c s of t h e different c l a s s e s of n o r m a t i v e consumption and i n v e s t m e n t models d i s c u s s e d in this study have been s u m m a r i z e d i n Table V-111 Since the headings of t h e v a r i o u s e n t r i e s ' s e e Kenneth A r r o w , " T h e Role of S e c u r i t i e s i n the Optimal Allocation of R i s k - B e a r i n g , " -Review of Economic Studies, A p r i l 1964 -m- - J a c k H i r s h l e i f e r , "Efficient Allocation of Capital i n a n U n c e r t a i n World, " The A m e r i c a = Economic Review, May 1964 J a c k H i r s h l e i f e r , "Investment Decision Under Uncertaintv: ChoiceT h e o r e t i c Approaches, " The Q u a r t e r l y J o u r n a l of E c o n o m i c s , November 1965 J a c k H i r s h l e i f e r , "Investment Decision Under Uncertainty: Applications of t h e S t a t e - P r e f e r e n c e Approach, I ' to a p p e a r i n The Q u a r t e r l y J o u r n a l of Economics a r e self - expl.a.natory, no f u r t h e r cornmefits beyond what h a s a l r e a d y been s a i d a p p e a r w a r r a n t e d Yes Yes Yes LongRun 3 Survival 3 Models I-IV Ch I1 Yes Yes Yes Yes No Not a p p l i c a b l e Yes No No No No Yes Yes No Yes No No Yes - Yes - No yes - Yes - No Yes No - - - No NO Yes Variable Discount * - In dUced NA Yes No Yes duced k- N NA In- - Yes - No Lnduced - - creasing In- Monotone duced duced Yes Yes Yes Yes Yes yes2 Yes Inputs In- Yes Yes Lnduced Strictly Concave Utility of Capital Yes No Yes Yes Yes yes2 Yes Yes Yes Yes Yes Yes Yes NO No More Than One Risky Productive Opportunities - Only i n m o d e l s of R a m s e y a n d of S a m u e l s o n a n d Solow Only i n m o d e l s of S a m u e l s o n and Solow and of Y a a r i , doos t o t a l n u m b e r of o p p o r t u n i t i e s e x c e e d 1 - Except i n m o d e l s of S a m u e l s o n a n d Solow a n d of Yaari, t h i s d e c i s i o n i s obtained a s a r e s i d u a l - Yes ChanceConstr NA Yes MeanVariance Yes Yes - Yes yes yes yes yes1 Yes yes - Yes Yes Yes Ye5 No Yes - No lity Function Constant Phelps 2 Model Type Fisher Predetermined Is Model Sequenltial? Consumption Decisions Strictly Concave, Monotone Consumption Investment Decisions Concern of Model No No No No No No yes3 Yes Nonconstant Returns To Scale NORMATIVE INVESTMENT AND CONSUMPTION MODELS: A COMPARATIVE SUMMARY T A B L E VlII Yes No Yes Yes 'Yes No yes2 Yes Financia1 Opportunities Yes Yes Yes Yes No Yes Yes Yes Wealth Yes No No Yes No Yes No Yes Yes Yes Yes Yes Yes Yes No Yes Yes No AlgorithNonmic Capior tal AnalyIntic come SoluStream tion BIBLIOGRAPHY J ~ c z g l ,V o r l e s u n g e n u b e r Funktionalgleichungen und I h r e Anwendungen, S t u t t g a r t , B i r k h a u s e r V e r l a g , 196 - J ~ c z &a n d S G o l a s s , Funktionalgleichungen d e r T h e o r i e l d e r G e o m e t r i s c h e n Obj ekte, W a r s a w , P a n s t w o w e Wydawnic -two Naukowe, 1940 M a u r i c e A l l a i s , "LIExte;.,sion d e s ~ h \ e o r i e s e l l E q u i l i b r e d Economique ~ n ; r a l e t du Rendement Social a u c i s du Risque," E c o n o r n e t r i c a , A p r i l 1953 Kenneth A r r o w B e r n o u l l i Utilitv I n d i c a t o r s f o r D i s t r i b u t i o n s Over A r b i t r a r y Spaces, T e c h n i c a l R e p o r t No 59, D e p a r t m e n t of E c o n o m i c s , Stanford U n i v e r s i t y , J u l y 1958 I Kenneth A r r o w , " C o m m e n t o n D u e s e n b e r r y i s ' The P o r t f o l i o A- -p r o a c h t o t h e Demand f o r Money a n d o t h e r A s s e t s ' ", p Review of Econ.omic s and Statis t i c s , Supplement, F e b r u a r y 1963 Kenneth A r r o w , " T h e Role of S e c u r i t i e s i n the: p t i m a l : A l l b c a t i o n of R i s k - b e a r i n g " , Review of E c o n o m i c Studies, A p r i l 1964 William E a u m o l , " A E x p e c t e d Gain-Confidence L i m i t C r i t e r ion f o r P o r t f o l i o Selection, " Management Science, October 1963 R i c h a r d Bellman, Adaptive C o n t r o l P r o c e s s e s , P r i ~ c e t o n , P r i n c e t o n U n i v e r s i t y P r e s s , 1961, Ric-hard B e l l m a n , .Cynarnl c P r o g r a m m i n g P r i n c e t o r , Pri-.lcetc- U ~ i v e r s i t y r e s s , 1957 P Da-lel 3ercou!.'i.i, " E x p o s i t i o n o i a New T h e o r y o n t h e M e a s u r e m e n t of Risk", transl L o u i s e S o m m e r , -c o ~ o m e t-s c a , E ! J a n u a r y 19 54 S U F H a r o l d B i e r m a a ~ l d P ~ ~ OSmidt, The C a p i t a l Budgeting Decis-ior, New York, Ma.cMilla.2, 1960 J Black, " O p t i m u m Savings R e c o n s i d e r e d , o r R a m s e y Without T e a r s " , -c- o m i c J o u r c a l , J u r e 1962 E- - on- P E u g e n von ~ t h m - ~ a w e r k ,o s i t i v e T h e o r y o f Capital, t r a n s l William S m a r t , New York, G E ~ t e c h e r t , 1923 K a r l B o r c h , " A Note on Utility and Attitudes to Risk", Ma - n a g e m e ~ tScience! 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HEMMING HAKANSSON T e diaclerta;tion of N i b Hemming Hakaneson is h appsoved, and it is acceptable In quality and form for publication on mlcrof''ilm: * r Committee Chairman Unlveraity of California,... unit of capital so invested (random variable) That is, if we invest an amount in i at the beginning of the period, we will obtain p Q at the end of that period (constant returns to 1.l scale)... '' at the beginning of the jth period (decision variable) - borrowing rate of interest - lending rate of interest, where rB rL > - transformation of capital invested in opportunity i in the jth