UBC Theses and Dissertations

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UBC Theses and Dissertations

Preferences, endowments and beliefs as revealed in market prices Sick, Gordon A. 1980

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PREFERENCES, ENDOWMENTS AND BELIEFS AS REVEALED IN MARKET PRICES by GORDON ARTHUR SICK B . S c , The U n i v e r s i t y o f C a l g a r y , 1971 M . S c , The U n i v e r s i t y of T o r o n t o , 1972 M . S c , The U n i v e r s i t y o f B r i t i s h - C o l u m b i a ( B u s i n e s s A d m i n i s t r a t i o n ) , 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES ( F a c u l t y of Commerce and.Business  Administration)  We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d  THE UNIVERSITY OF BRITISH COLUMBIA November 1980 ©  Gordon A r t h u r S i c k ,  1980  In  presenting this  thesis  an advanced degree at  further  for  of  the  requirements  freely  available  for  this  this  thesis for  financial  The  Vtv^lON  gain s h a l l  Date  Of  not  ft/jAt/cj  U n i v e r s i t y o f B r i t i s h Columbia  2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  or  i s understood that c o p y i n g or p u b l i c a t i o n  written permission.  hh3parl.mer1l.-wf  that  thesis  s c h o l a r l y purposes may be granted by the Head of my Department It  fo  r e f e r e n c e and study.  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f  by h i s r e p r e s e n t a t i v e s . of  fulfilment  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make i t I  in p a r t i a l  be allowed without my  ii  Abstract T h i s t h e s i s examines c o n d i t i o n s under which p r i c e s s i g n a l information  about a g e n t s ' p r e f e r e n c e s , endowments a n d / o r  information. because i t  In a m u l t i - p e r i o d  economy, t h i s  information  is  important  h e l p s agents t o make i n f e r e n c e s about f u t u r e p r i c e s .  In a s i n g l e p e r i o d economy, t h i s are only i n t e r e s t e d it  probability  i s important  i s i m p o r t a n t b e c a u s e , even i f  in other agents' p r o b a b i l i t y  agents  information,  f o r them to be a b l e to d i s t i n g u i s h i t s e f f e c t  on  p r i c e s from the e f f e c t s o f p r e f e r e n c e s and endowments on p r i c e s . S e v e r a l exchange economy models a r e c o n s t r u c t e d to see under what conditions a f u l l y  informing r a t i o n a l  expectations  (FRE) e x i s t s i n which the r e l e v a n t i n f o r m a t i o n prices. setting  equilibrium  i s r e v e a l e d by  One c l a s s o f models i s i n a two p e r i o d s t a t e i n which p r e f e r e n c e s e x h i b i t  aggregate p r e f e r e n c e s e x i s t ) .  It  l i n e a r r i s k t o l e r a n c e (so t h a t  i s shown t h a t a FRE e x i s t s  r e v e a l s aggregate p r e f e r e n c e p a r a m e t e r s .  In another  p e r i o d s t a t e p r e f e r e n c e model w i t h power u t i l i t y p r e f e r e n c e s do n o t e x i s t ) ,  it  can r e v e a l l o c a l i n f o r m a t i o n endowments.  preference  that  two  ( i n which  aggregate  i s shown t h a t p r i c e s  generically  about the d i s t r i b u t i o n  of agents'  In a n o t h e r c l a s s o f m o d e l s , i n both one p e r i o d and  two p e r i o d s e t t i n g s w i t h s p e c i f i c d i s t r i b u t i o n a l (normal and n o n - c e n t r a l  gamma r e t u r n s ) ,  under which p r i c e s r e v e a l  information  assumptions  c o n d i t i o n s a r e found  about  probabilities,  aggregate r i s k p r e f e r e n c e s and aggregate i m p a t i e n c e .  ,  i i i  The t h e s i s d i s c u s s e s the n o t i o n o f a r a t i o n a l e x p e c t a t i o n s e q u i l i b r i u m as a s o l u t i o n o f a f i x e d p o i n t p r o b l e m .  It  also discusses information  terms o f a - a l g e b r a s and p a r t i t i o n s o f s t a t e s p a c e s .  in  iv TABLE OF CONTENTS  Page ABSTRACT  ii  LIST OF TABLES . .  vii  LIST OF FIGURES  vi i i  ACKNOWLEDGEMENTS  ix  Chapter  1.  OVERVIEW  1  I n f o r m a t i o n and p r i c e s  1  Overview o f the f o l l o w i n g 2.  chapters  3  THE RATIONAL EXPECTATIONS CONCEPT  7  Introduction Information,  7 b e l i e f s and r a t i o n a l  Existence of rational  expectations  7  expectations e q u i l i b r i a  Other papers c o n c e r n i n g r a t i o n a l  13  expectations  19  Footnotes 3.  25  REVELATION OF AGGREGATE PREFERENCE PARAMETERS  28  Introduction  28  Notation  30  A power u t i l i t y economy  33  Intermediate  39  l a b o r income  FRE's w i t h other l i n e a r r i s k t o l e r a n c e u t i l i t y c l a s s e s Rollover of portfolios  at date 1  . . . .  44 46  V  Chapter  Page A model i n which d a t e 0 p r i c e s r e v e a l aggregate p r e f e r e n c e s and agents r e - b a l a n c e p o r t f o l i o s  4.  i n the i n t e r m e d i a t e p e r i o d . . .  50  Conclusion  55  Footnotes .  56  REVELATION OF INDIVIDUAL ENDOWMENTS Introduction  . . . .  61  '  61  Market s t r u c t u r e , n o t a t i o n and p r e f e r e n c e s The r e v e l a t i o n o f i n f o r m a t i o n  68  Conclusion  79  Footnotes  81  Appendix:  D e t a i l s o f the counterexample p r o v i d i n g a J a c o b i a n o f  l e s s than f u l l 5.  .63  rank  83  PRICES REVEALING AGGREGATE RISK AVERSION, IMPATIENCE AND PROBABILITY BELIEFS  85  Introduction  85  N o t a t i o n and s e t t i n g One p e r i o d m o d e l s : One p e r i o d m o d e l :  .  89  Exponential u t i l i t y Normally d i s t r i b u t e d  91 returns  One p e r i o d m o d e l s :  Gamma d i s t r i b u t e d  returns  One p e r i o d m o d e l s :  Other d i s t r i b u t i o n s  Two p e r i o d m o d e l s :  Exponential u t i l i t y  Two p e r i o d m o d e l s :  Normally d i s t r i b u t e d r e t u r n s  Two p e r i o d m o d e l s :  Gamma d i s t r i b u t e d r e t u r n s  Two p e r i o d m o d e l s :  Portfolio rollovers  93  . .  and u t i l i t y c l a s s e s  96 . .  101 104  .  108 Ill 118  Conclusion  119  Footnotes  120  vi Chapter 6.  Page  CONCLUDING :• REMARKS What i s a r a t i o n a l  123 expectations equilibrium?  124  Do p r i c e s r e v e a l i n f o r m a t i o n about p r e f e r e n c e s ? Do p r i c e s r e v e a l i n f o r m a t i o n about the d i s t r i b u t i o n  125 of  a g e n t s ' endowments?  125  Do p r i c e s r e v e a l and d i s t i n g u i s h between p r e f e r e n c e and probability  parameter i n f o r m a t i o n ?  125  Conclusion  127  BIBLIOGRAPHY  128  vii LIST OF TABLES Table 5.1  f^e C o n d i t i o n s f o r a FRE i n Theorem 5.7  • •  j  1 1 7  LIST OF FIGURES Figure  Page  1.1  Information  feedback  2  2.1  A.family of functions  3.1  Edgeworth box w i t h l i n e a r c o n t r a c t c u r v e , homothetic  15 !  i n d i f f e r e n c e c u r v e s and endowment e  57  3.2  I n t e r i o r optimum  .  4.1  Market regime M  65  4.2  Market regime M'  65  4.3  Market regime M"  5.1  S o l u t i o n of Equation (5.18)  94  5.2  Graphs o f the r i g h t hand s i d e o f E q u a t i o n ( 5 . 2 2 )  98  5.3  Graphs o f F ( e ) , the r i g h t hand s i d e o f E q u a t i o n ( 5 . 2 3 )  98  5.4  Graphs o f F ( e ) , the r i g h t hand s i d e o f ( 5 . 4 4 )  5.5  Graphs o f G ( e ) , the r i g h t hand s i d e o f ( 5 . 4 5 )  5.6  Graphs o f G ( e ) , the r i g h t hand s i d e o f ( 5 . 4 5 ) f o r a i < a  . .  59  65  113 f o r a >cx x  2  2  . . . .  115  . . . .  115  ix ACKNOWLEDGEMENTS I would l i k e to thank the f o l l o w i n g  people f o r v a r i o u s h e l p f u l  and s u g g e s t i o n s , w h i l e a b s o l v i n g them o f any r e s p o n s i b i l i t y f o r which a r e s o l e l y m i n e :  comments  errors,  Don Brown, F i s c h e r B l a c k , Michael B r e n n a n , Erwin  D i e w e r t , S a n f o r d Grossman, Robert C. M e r t o n , Paul Milgrom and Stephen R o s s . Above a l l , I would l i k e to thank my a d v i s e r , A l a n K r a u s , f o r  suggesting  the g e n e r a l t o p i c c o n s i d e r e d h e r e , as w e l l as f o r p r o v i d i n g many hours o f f r u i t f u l d i s c u s s i o n on both broad and f i n e p o i n t s c o n t a i n e d h e r e i n . F i n a n c i a l s u p p o r t was p r o v i d e d by t h e Canada C o u n c i l , the S o c i a l S c i e n c e s and Humanities Research C o u n c i l o f Canada, and a g r a n t t o Y a l e U n i v e r s i t y by the Dean W i t t e r  Foundation.  1. Chapter 1  Overview  I n f o r m a t i o n and p r i c e s In a dynamic economy, agents w i l l for financial will  a s s e t s and consumption goods, a n d , a t any d a t e ,  be u n c e r t a i n about what p r i c e s w i l l  at future dates. uncertanty  prevail  i n those markets  The u n c e r t a i n t y about f u t u r e p r i c e s stems from  about exogenous p a r a m e t e r s , such as p r o d u c t i v i t y  the w e a t h e r ) , w a r , a r r i v a l etc.  f a c e a sequence o f markets  o f new t e c h n o l o g y , p o p u l a t i o n  V a r i o u s agents o f t e n have more i n f o r m a t i o n  parameters than o t h e r s , o r agents may s i m p l y have information  (or  shifts,  about these different  b e c a u s e , f o r example, two e l e c t r o n i c s . . e x p e r t s may  o b s e r v e t e c h n o l o g y breakthroughs  in different  product  markets.  A t any p o i n t i n time these types o f u n c e r t a i n t y may be c l a s s i f i e d i n t o two c a t e g o r i e s . world w i l l 1981").  The o t h e r c a t e g o r y r e l a t e s to the parameters o f the p r o b a -  "expected r a i n f a l l  o f what s t a t e s o f the w o r l d w i l l i n Texas i n 1 9 8 1 " ) .  agents have the same i n f o r m a t i o n  only  to d i f f e r e n t i a l  o f the p r o b a b i l i t y  distribution  occur  For s i m p l i c i t y ,  information  (e.g.,  suppose  and b e l i e f s about what  s t a t e s o f the w o r l d c o u l d o c c u r and t h a t d i f f e r e n t i a l pertains  the  o c c u r i n the f u t u r e ( e . g . , " s e v e r e drought i n Texas i n  b i l i t y distribution  that a l l  One c a t e g o r y r e l a t e s to what s t a t e s o f  about the  ("probability  information parameters  parameters").  U n c e r t a i n t y about f u t u r e p r i c e s may a l s o stem from  uncertainty  about a g e n t s ' p r e f e r e n c e p a r a m e t e r s , such as r i s k a v e r s i o n , i m p a t i e n c e  2. and the functional c l a s s of agents' u t i l i t y functions (e.g., exponential, l o g a r i t h m i c ) . A l s o , i f preferences are not such that an aggregate investor e x i s t s , the j o i n t d i s t r i b u t i o n of wealth endowments and preferences also a f f e c t s (future) p r i c e s . C e r t a i n l y agents have better information about t h e i r own  pre-  ferences than they have about other agents' preferences. Agents'rdemands i n current markets generally depend upon the information that they have about p r o b a b i l i t y parameters, preferences and endowments.  Thus current prices are a function  of a l l of t h i s information and hence r e f l e c t , at l e a s t to some extent, t h e i r information.  Agents may  then use the endogenous  information impounded i n prices to r e f i n e t h e i r own exogenous information.  The improved information a l t e r s agents' demands  and thus a l t e r s the way prices depend upon, and hence r e f l e c t , a l l of the agents' information i n the f i r s t place.  Information,  demands, and prices are thus involved i n a feedback loop, as depicted i n Figure information  1.1. —->  demands  : Figure 1.1  >- prices  i  Information feedback  Because of the feedback, the market may  f a i l to e q u i l i b r a t e .  Problems of t h i s s o r t are discussed i n Chapter 2. The assumption that agents use a l l a v a i l a b l e information, including the endogenous information impounded i n prices i s an  3. assumption about the r a t i o n a l i t y of agents.  Because of t h i s ,  the branch of economics dealing with these issues i s c a l l e d the "theory of r a t i o n a l expectations."  As discussed i n Chapter 2,  t h i s involves a confusion between expectations (or b e l i e f s ) and information, so that the t i t l e i s r e a l l y a misnomer.  However,  we s h a l l r e f e r to these issues as being about r a t i o n a l  expecta-  t i o n s , i n accordance with the usage i n the l i t e r a t u r e . The r a t i o n a l expectations l i t e r a t u r e , which i s reviewed b r i e f l y i n Chapter 2, has generally only d e a l t with cases where prices convey information about p r o b a b i l i t y parameters, overlooking preference and endowment parameters,  t h i s t h e s i s examines  models where prices convey information about preferences, endowments, and j o i n t l y about preferences and  probabilities.  Overview of the f o l l o w i n g chapters Chapter 2 provides a general discussion of information and d i s t i n g u i s h e s information from b e l i e f s .  I t a l s o defines a r a t i o n a l  expectations e q u i l i b r i u m as an e q u i l i b r i u m i n an economy where agents use t h e i r own exogenous information, as well as the endogenous information impounded i n p r i c e s .  Existence of a r a t i o n a l  expectations e q u i l i b r i u m i s characterized as a f i x e d point problem i n a f u n c t i o n space.  Two types of existence problems are discussed,  which motivate i n t e r e s t i n studying the existence of f u l l y informing  4..  rational  expectations e q u i l i b r i a  with a b r i e f literature  (FRE's).  The c h a p t e r c o n c l u d e s  s u r v e y o f some o f the m i c r o - and macro-economic  on r a t i o n a l  expectations.  Chapter 3 models s e v e r a l two p e r i o d s t a t e p r e f e r e n c e economies i n which aggregate p r e f e r e n c e s e x i s t power o r l o g u t i l i t y , o r e x p o n e n t i a l  0  (agents have extended  utility).  For such e c o n o m i e s ,  market p r i c e s a t c u r r e n t and f u t u r e dates depend on an aggregate r i s k a v e r s i o n p a r a m e t e r , so t h a t c u r r e n t p r i c e s can r e f l e c t  the  v a l u e o f the p r e f e r e n c e p a r a m e t e r , and hence h e l p to r e s o l v e some o f the u n c e r t a i n t y are shown to e x i s t .  about f u t u r e p r i c e s .  In such m a r k e t s ,  The s t r o n g assumptions made about p r e f e r e n c e s  and market s t r u c t u r e e n a b l e i n v e s t o r s to s i m p l i f y portfolio  FRE's  their  initial  c h o i c e , however, i n such a way t h a t they can form  optimal  demands i n the c u r r e n t m a r k e t , w i t h o u t h a v i n g to a s s e s s the  market's  aggregate r i s k a v e r s i o n parameter.  A model i s p r e s e n t e d i n  the  Appendix t o Chapter 3 i n which aggregate p r e f e r e n c e s e x i s t ,  but  no such s i m p l i f i c a t i o n  i s p o s s i b l e f o r a g e n t s , f o r c i n g them to  i n f e r aggregate r i s k a v e r s i o n from c u r r e n t  prices.  Chapter 4 s t u d i e s the q u e s t i o n o f whether p r i c e s can r e v e a l information  about a g e n t s ' endowments i n a two p e r i o d w o r l d .  When  t h e r e a r e I a g e n t s , i n g e n e r a l t h e r e must be a t l e a s t I p r i c e s convey the i n f o r m a t i o n  about endowments.  The problem o f e s t a b l i s h -  i n g the e x i s t e n c e o f F R E ' s t h a t convey m u l t i v a r i a t e more d i f f i c u l t than t h a t o f a n a l y s i n g the u n i v a r i a t e ter 3.  to  information  is  F R E ' s o f Chap-  In t h i s case the r e s u l t s are not c o n s t r u c t i v e and e s t a b l i s h  5.  generic existence only.  That i s , i f  economy, a " s m a l l p e r t u r b a t i o n " if  an economy has a F R E , a l l  FRE's.  a FRE does not e x i s t f o r an  o f the economy y i e l d s a FRE, a n d ,  "sufficiently  c l o s e " economies have  The g e n e r i c e x i s t e n c e o f F R E s i n o n e - p e r i o d economies, 1  where p r o b a b i l i t y  information  i s unknown, has been e s t a b l i s h e d  under c e r t a i n s t r o n g d i m e n s i o n a l i t y assumptions by A l l e n 1979],  She uses d i f f e r e n t i a l  [1978,  t o p o l o g y to g e t these r e s u l t s .  r e s u l t s i n C h a p t e r 4 s t u d y the e x i s t e n c e o f l o c a l l y f u l l y rational  expectations e q u i l i b r i a  (LFRE's) o n l y :  if  The  informing  the v e c t o r  endowments i s c o n f i n e d to a s u i t a b l y s m a l l s e t , p r i c e s r e v e a l endowment v e c t o r i n e q u i l i b r i u m .  By o n l y s e a r c h i n g f o r  of the  LFRE's,  Chapter 4 e s t a b l i s h e s g e n e r i c e x i s t e n c e under s l i g h t l y weaker d i m e n s i o n a l i t y assumptions than A l l e n , by u s i n g o n l y a n a l y t i c r e s u l t s about i m p l i c i t  well-known  functions.  In C h a p t e r 5 , one and two p e r i o d economies w i t h one r i s k y a s s e t are s t u d i e d to see when p r i c e s can r e v e a l i n f o r m a t i o n  about  probability  risk  parameters and p r e f e r e n c e parameters (aggregate  a v e r s i o n , a n d , i n the two p e r i o d economy, i m p a t i e n c e ) . to the r e s u l t s o f Chapter 4 , these m u l t i v a r i a t e are c o n s t r u c t i v e . they g i v e  exponential  information  The r e s u l t s are s t r o n g e r than g e n e r i c  sufficient,  a FRE t o e x i s t .  In  contrast results  results:  and i n some c a s e s , n e c e s s a r y c o n d i t i o n s  The models a r e based upon i n t e r t e m p o r a l l y  u t i l i t y f u n c t i o n s and i n v o l v e two s p e c i f i c  for  additive  probability  f a m i l i e s f o r the increments to the s o c i a l w e a l t h p r o c e s s :  normal  6.  and n o n - c e n t r a l gamma. typically  The c o n d i t i o n s f o r the e x i s t e n c e o f a FRE  i n v o l v e the r e l a t i o n s h i p between a g e n t s ' r a t e o f  impatience  and d i s c o u n t bond p r i c e s ( i n the one p e r i o d models) and the s l o p e o f the term s t r u c t u r e o f i n t e r e s t r a t e s  ( i n the two p e r i o d m o d e l s ) .  Chapter 6 p r o v i d e s some c o n c l u d i n g r e m a r k s .  7.  Chapter 2  The Rational Expectations Concept  Introduction This chapter provides a general d i s c u s s i o n of r a t i o n a l expectations models, i n terms of t h e i r d e f i n i t i o n , existence and general properties.  I t s t a r t s with a general d i s c u s s i o n of information, i n  terms of a-algebras and p a r t i t i o n s i n p r o b a b i l i t y spaces, and proceeds to a c h a r a c t e r i z a t i o n of r a t i o n a l expectations e q u i l i b r i a as f i x e d points ( i n a Banach space).  C o n t i n u i t y and m e a s u r a b i l i t y  problems that create problems f o r the existence of r a t i o n a l expectations e q u i l i b r i a are discussed.  This leads to a study of the  existence of f u l l y informing r a t i o n a l expectations e q u i l i b r i a (FRE's), the existence of which i s more r e a d i l y v e r i f i a b l e than the existence of general r a t i o n a l expectations e q u i l i b r i a . The chapter concludes with a survey of some of the r a t i o n a l  ex-  pectations models i n the l i t e r a t u r e . Information, b e l i e f s and r a t i o n a l expectations Consider a market system f o r S goods i n v o l v i n g uncertainty about, say, f u t u r e endogenous or exogenous production l e v e l s . (i=l,  I) receives information.A  certain variables.  One may  about the true value of the un-  think of t h i s i n terms of a p r o b a b i l i t y  s p a c e ^ (ft, 8, p ), with s t a t e space a, measure P^.  Suppose agent i  a-algebra B and  probability  Agent i ' s b e l i e f s are represented by P.. and his informa-  tion A.j i s a sub-a-algebra of B upon which he may 2) expectations. •  take c o n d i t i o n a l  8. many elements ( s t a t e s ) , B  I f t h e s t a t e space a has f i n i t e l y  i s t h e c o l l e c t i o n o f a l l s u b s e t s o f Q ( B = 2 ) , and P.. i s g e n e r a t e d fi  by t h e n o n - n e g a t i v e p r o b a b i l i t y  o f each s t a t e .  t i o n A.j may be r e g a r d e d as a p a r t i t i o n  Agent i ' s  informa-  o f o, ( a c t u a l l y t h e p a r t i t i o n  g e n e r a t e s A..) such t h a t agent i can d i s t i n g u i s h which member o f the p a r t i t i o n  has o c c u r r e d , b u t cannot d i s t i n g u i s h which s t a t e i n  the p a r t i t i o n  member has o c c u r r e d .  F o r example, i f n = {1  suppose A., i s g e n e r a t e d by t h e p a r t i t i o n  {{1,2>, {3},  n},  {4,...,n}}.  Then, i f s t a t e 1 o c c u r s , agent i o n l y knows t h a t e i t h e r s t a t e 1 o r state 2 occurred.  C o n d i t i o n i n g on a p a r t i t i o n  f o r m i n g t h e usual c o n d i t i o n a l  member B  involves  probabilities.  P(tj}|B)  =  P  (  j  g  p  j  k  (jeB).  )  keB  The most i m p o r t a n t  conditional  to g e n e r a t e demand f u n c t i o n s . p r i c e s , where £ Let  (co) e  +  e x p e c t a t i o n s a r e t h o s e used  L e t A CE  be a f e a s i b l e s e t o f  +  i s the non-negative orthant i n Euclidean n-space.  be agent i s endowment a t co  e  n .  The - excess demand  function: ^ : A x n x { A . C 2 | A . i s a s u b - a - a l g e b r a o f B} -> £ N  s o l v e s , f o r each mtt and i n f o r m a t i o n A^ t h e e x p e c t e d maximization problem.  (2.1)  utility  That i s ,  ^(p^.A.)  = a r g max  E^U^x+x^co))^.).  {x e & | p ' = 0 } s  x  S  H e r e , " a r g max" r e f e r s to the m a x i m i z i n g argument o f a m a x i m i z a t i o n problem and  i s von Neumann-Morgenstern u t i l i t y .  For s i m p l i c i t y ,  we c o n s i d e r o n l y s i n g l e - v a l u e d demands, w i t h n o n - s a t i a t i o n . i s a n o n - c o n s t a n t f u n c t i o n o f u , we r e q u i r e i t  If  x\(to)  t o be an A . - m e a s u r a b l e  random v a r i a b l e , so t h a t agent i knows h i s budget c o n s t r a i n t .  If  c  utility is sufficiently  smooth, e t c . then C j ( p , » , A . . )  be a p r o p e r ( A ^ - m e a s u r a b l e ) random v a r i a b l e f o r a l l a f i n i t e s t a t e s p a c e , t h i s means t h a t o v e r members o f the p a r t i t i o n  : Q, •> & p, A , . .  (p,. A.) will / 5  If n i s  be c o n s t a n t  g e n e r a t i n g A . . , s i n c e agent i  d i s t i n g u i s h between s t a t e s i n the same member o f the  will  cannot  partition  g e n e r a t i ng A ^ . L o o s e l y s p e a k i n g , f o r each r e a l i z a t i o n u o f the  probability  s t a t e space Q, agent i r e c e i v e s some (but perhaps not a l l ) about the v a l u e o f co (he can t e l l  information  whether o r not us i s i n any g i v e n  e v e n t i n A.., b u t not whether to i s o r i s n o t i n any s m a l l e r e v e n t s ) , as w e l l as the v a l u e o f h i s endowment. h i s (random) excess demand f u n c t i o n  He then s o l v e s (2.1) f o r  ^(p,w,A.).  F i x w and impose the market c l e a r i n g c o n d i t i o n z. t ; ( p , t o , A . ) = 0 .  (2.2);  i  1  T h i s d e f i n e s the p r i c e p, a s s u m i n g , f o r s i m p l i c i t y , t h a t prices are unique.  S e t P(W) = p.  market c l e a r i n g p r i c e f u n c t i o n p". tions, p will variable.  equilibrium  Repeat t h i s f o r a l l co t o g e t the Under a p p r o p r i a t e r e g u l a r i t y  condi-  be a 8-measurable f u n c t i o n o f o> and hence a t r u e random  I n d e e d , s i n c e p can o n l y vary a c c o r d i n g to the  information  { A . . } communicated to the a g e n t s , p s h o u l d be A - m e a s u r a b l e where = A-^vA^v..  A If  . v A j i s the s m a l l e s t a - a l g e b r a c o n t a i n i n g a l l  is finite,  partitions  A c o r r e s p o n d s to the common r e f i n e m e n t o f a l l  g e n e r a t i n g the A . j ' s .  endowment" o f  In some s e n s e , A i s t h e " s o c i a l  p o i n t i s t h a t p = P(OJ) v a r i e s w i t h the  mation t h a t ( o t h e r ) i n d i v i d u a l s r e c e i v e .  individual  L e t P ( A ) denote the a - a l g e b r a g e n e r a t e d byp (as d e f i n e d - 1  in footnote 2 . , i . e . ,  the i n f o r m a t i o n conveyed by t h e p r i c e  function  A.j VP~^(A) be the s m a l l e s t a - a l g e b r a c o n t a i n i n g both A., and  p"^(A), all  A rational  infor-  i n c l u d e the i n f o r m a t i o n conveyed by P i n s e l e c t i n g h i s demand  function.  Let  the  information.  The i m p o r t a n t  will  the A . j ' s .  i.e.,  all  the i n f o r m a t i o n c o n t a i n e d i n A ^ and P.  the i n f o r m a t i o n a v a i l a b l e to agent  '  This  is  i.  T h i s y i e l d s the f o l l o w i n g analogue o f the tatonnement p r o c e s s .  A rational  individual will  now have f o r each co, e x c e s s demand a t  p r i c e p of 5^(p^.A^vP ( A ) ) . -1  Note t h a t now, the random excess d e c  mand 5^ depends on the whole p r i c e f u n c t i o n P : fi-*- & , +  since .it  communicates i n f o r m a t i o n , as w e l l as on the p r i c e p, s i n c e i t mines t h e budget c o n s t r a i n t .  deter-  A l s o , £ . depends on co o n l y through  the  i n f o r m a t i o n conveyed by A ^ v p ' ^ A ) , so t h a t under a p p r o p r i a t e r e g u - ; l a r i t y conditions £.(p,•,A^vp~^(A)) Now,  fix  i s A.vp~^(A)-measurable.  co and impose the market c l e a r i n g  condition  (2.3) Z.5.( ,(o,A.vp P  (A))=  0 -\  11.  p'(co)  Set  function will  = p. p  1  :R  Repeating t h i s f o r a l l  coeft  d e f i n e s a new p r i c e  w h i c h , under a p p r o p r i a t e r e g u l a r i t y  A  be A - m e a s u r a b l e ( i . e . ,  random).  conditions  Of c o u r s e , s i n c e agents have  f o r m u l a t e d t h e i r demands w i t h r e s p e c t to a new i n f o r m a t i o n (A^vp"^(A),  r a t h e r than  g e n e r a l , be d i f f e r e n t  A..),  the new p r i c e f u n c t i o n  from p.  expectations using A v p ' ^ ( A . )  p'  set  will,  in  I n d i v i d u a l s must now r e - r e v i s e  their  r a t h e r than A . i n ( 2 . 1 ) .  the  With  market c l e a r i n g c o n d i t i o n , t h i s g i v e s a new p r i c e f u n c t i o n p", so on ad i n f i n i t u m .  The q u e s t i o n o f whether such an analogue o f  the tatonnement p r o c e s s w i l l  a c t u a l l y converge i s , o f c o u r s e , q u i t e  d i f f i c u l t and w i l l  n o t be a n a l y s e d h e r e .  that a fixed point  (if  is called a rational A rational  it  exits)  expectations  o f such a sequence o f  (For  point  s  (random v a r i a b l e )  re-adjustments  ? ( (o ),co,A vp" (A)) = 0 i  P  J  i  p : n--:-*  &  +  such t h a t  .  m a t h e m a t i c a l r i g o u r , add an a p p r o p r i a t e s p r i n k l i n g o f  surely.")  exists  then 1  i  is  equilibrium.  e x c e s s demands a r e d e f i n e d by ( 2 . 1 )  (2.4)  The i m p o r t a n t  e x p e c t a t i o n s e q u i l i b r i u m o c c u r s when t h e r e  an A - m e a s u r a b l e p r i c e f u n c t i o n if  and  "almost  T h u s , agent i uses h i s own exogenous i n f o r m a t i o n A., and  the endogenous i n f o r m a t i o n  P~^(A) conveyed by p r i c e s to form random  excess demand £ . (p (•) ,• . A . - v p  -1  ( A ) ) , which i s A..vp~^ ( A ) - m e a s u r a b l e .  Note t h a t the p r i c e f u n c t i o n p which agents assume h o l d s formulating  t h e i r e x p e c t a t i o n s and demands i s a c t u a l l y the one  t h a t c l e a r s the m a r k e t . a rational  in  T h i s i s the " s e l f - f u l f i l l i n g "  expectations e q i l i b r i u m .  nature  of  T h u s , the c o n s i s t e n c y t h a t  12. is implicit  in s e l f - f u l f i l l e d  " e x p e c t a t i o n s " i s not r e a l l y one  of b e l i e f s or expectations, for (P - d i f f e r e n t  t h e r e c o u l d be d i f f e r e n t i a l  from P^) o r , even i n c o r r e c t b e l i e f s  n  from N a t u r e ' s " t r u e "  probability  law P ) .  (P^  Rational  r e a l l y o n l y imposes a c o n s i s t e n c y on a g e n t s '  beliefs.  different  expectations  information  A^VP"^(A).  R a t i o n a l e x p e c t a t i o n s r e q u i r e t h a t agents o n l y f o r e c a s t the  possi-  b i l i t y o f f e a s i b l e e v e n t s — t h a t i s , . t h e y - u s e the s t a t e space ft, and have a a - a l g e b r a A..VP~^(A;).  C  rectly  use a l l  B.  It  a l s o r e q u i r e s t h a t agents  a v a i l a b l e information  (as i n  A-  and  corBut,  p~^(A)).  does not r e q u i r e any homogeneity o r c o r r e c t n e s s o f p r o b a b i l i t y  beliefs  1  P.j, which a r e exogenous  it  t o . t h e model. In a r e p e a t e d economy, agents  may update P.. i n a B a y e s i a n manner, and o v e r t i m e , P^ may converge to n a t u r e ' s  "true"  P.  To make r a t i o n a l  empirically testable implications,  it  e x p e c t a t i o n s models y i e l d  o f t e n seems n e c e s s a r y to add  the h y p o t h e s i s t h a t b e l i e f s a r e homogeneous and " c o r r e c t " with P ) .  Moreover, i f  adjust for  this  agents have d i f f e r e n t i a l  i n order  a c t i o n s o r from p r i c e s .  to i n f e r  information  b e l i e f s , they must from o t h e r  For e x a m p l e , suppose a s u p e r i o r  a n a l y s t ( c h a r a c t e r i z e d by a l a r g e s - a l g e b r a A . , o r a v e r y partition)  i s a l s o an i n c u r a b l e o p t i m i s t  (coincident  agents' stock refined  and can be o n l y o b s e r v e d to  be " b u y i n g s t o c k " o r " b u y i n g l o t s o f s t o c k . "  A less  well-informed  a g e n t , b u t w i t h somewhat more " c o r r e c t " and p e s s i m i s t i c b e l i e f s , find i t  optimal  to s e l l s t o c k s when the a n a l y s t m e r e l y buys,  buy s t o c k s when the a n a l y s t buys l o t s o f  stock.  will  and,to  13.  C l e a r l y , agents may have e i t h e r homogeneous o r  differential  b e l i e f s w h i l e having e i t h e r homogeneous o r d i f f e r e n t i a l  information.  R a t i o n a l e x p e c t a t i o n s i s a misnomer f o r t h i s t h e o r y i n t h a t i t r e a l l y o n l y a t h e o r y o f the r a t i o n a l  use o f i n f o r m a t i o n ,  not  is of  beliefs or expectations. Having opened up the p o s s i b i l i t y o f r a t i o n a l  expectations  e q u i l i b r i a w i t h heterogeneous b e l i e f s , i n what f o l l o w s , we w i l l assume, f o r s i m p l i c i t y ,  t h a t agents have homogeneous b e l i e f s ,  s i n c e assuming heterogeneous b e l i e f s adds many parameters to models and o f t e n makes c l o s e d form s o l u t i o n s i m p o s s i b l e to a t t a i n . geneous b e l i e f s merely c l o u d the problem o f s t u d y i n g  Hetero-  differential  information. Existence of rational  expectations  equilibria  With t h i s machinery i n p l a c e , one can a p p r e c i a t e the  diffi-  c u l t i e s a s s o c i a t e d w i t h e s t a b l i s h i n g the e x i s t e n c e o f a r a t i o n a l e x p e c t a t i o n s e q u i l i b r i u m , much l e s s o f a n a l y z i n g problems o f or dynamics. information,  In s t a n d a r d models o f u n c e r t a i n t y w i t h o u t we have . A . , . = A ( \ / i )  = A^, = A ( \ / i ) ) .  differential  (and A = B i n the c e r t a i n t y  so t h a t the p r i c e f u n c t i o n conveys no i n f o r m a t i o n  stability  case),  ( i . e . , A^vp~^(A)  E x i s t e n c e o f e q u i l i b r i u m i s e s t a b l i s h e d by s t a n d a r d  methods, such as i n Debreu (1959) o r Arrow and Hahn ( 1 9 7 1 ) .  In  t h i s c a s e , p r i c e p i s a f i x e d p o i n t o f a c o n t i n u o u s mapping o f A into i t s e l f ,  w i t h the p r o p e r t y  t h a t a f i x e d p o i n t c o r r e s p o n d s to a  p o i n t o f zero aggregate excess demand.  For t h e g e n e r a l r a t i o n a l  e x p e c t a t i o n s problem w i t h  differential  i n f o r m a t i o n , we need a whole p r i c e random v a r i a b l e P :  A .  That  i s , we d e s i r e a f i x e d p o i n t o f some mapping i n a f u n c t i o n space o f random v a r i a b l e s , r a t h e r than j u s t a f i x e d p o i n t i n E u c l i d e a n S - s p a c e . T h i s f u n c t i o n s p a c e , endowed w i t h an a p p r o p r i a t e t o p o l o g y , w i l l Banach s p a c e .  This, in i t s e l f ,  i s no p r o b l e m , f o r t h e r e a r e f i x e d  p o i n t theorems f o r Banach s p a c e s .  The problem i s t h a t i t may n o t be  p o s s i b l e t o p o s t u l a t e enough c o n t i n u i t y , logical  be a  compactness and o t h e r  topo-  p r o p e r t i e s on t h e exogenous p a r t s o f t h e problem t o ensure  c o n t i n u i t y o f any u s e f u l mapping i n t h e Banach space o f p r i c e random variables. S.j(p,  T h i s o c c u r s because t h e e x c e s s demand random v a r i a b l e s  *,A.jVP  ^(A))  are n o t , i n g e n e r a l , continuous f u n c t i o n s ,  any a p p r o p r i a t e t o p o l o g y , o f t h e p r i c e random v a r i a b l e P.  under  That i s ,  i n g e n e r a l , t h e i n f o r m a t i o n communicated by p r i c e s , P~^(A) i s n o t , i n some s e n s e , a c o n t i n u o u s f u n c t i o n o f p. To i l l u s t r a t e  t h i s , c o n s i d e r a f a m i l y o f f u n c t i o n s on  {to : co>0}, p a r a m e t e r i z e d by 6e [0,1], and d e f i n e d by /  Suppose t h a t agents a r e a t t e m p t i n g  6  t o l e a r n something about ~<o  by o b s e r v i n g P g ^ ) . The i n v e r t i b i l i t y monotonicity are four  l-Q  of p  0  depends on i t s  and hence on t h e z e r o s o f i t s d e r i v a t i v e s .  There  cases:  1. 6e  [0,1/5).  f o r u>. 0 .  There a r e no s t a t i o n a r y p o i n t s f o r P ( w ) e  2.  e = 1/5.  3.  0  (1/5, 1/2).  £  and 4.  = 0 o n l y when «D = 0 .  Pg M  e  e  = 0 f o r e x a c t l y one .u* > 0 ,  P 'H 0  °° as o + 1 / 2 " .  [1/2,1].  p  ' ( c o ) * 0 f o r a l l co > 0 .  T h i s i s graphed i n F i g u r e 2 . 1 . F o r 0>. ^,P  ( w ) i s an i n v e r t i b l e  9  6  i t i s not i n v e r t i b l e .  <  p  ( i) u  f  l  = P (to ) = P, fl  2  Call  u  f u n c t i o n o f to, b u t f o r  i and 2 c o n f o u n d i n g  s a y . Since the s t a t i o n a r y point  goes t o + <>° as e -> 1 / 2 " , t h e r e a r e a r b i t r a r i l y pairs  co  l s  u> w i t h 2  arbitrarily different  Ico^-u) |' 2  c l o s e t o 1.  arbitrarily  w  large  l a r g e and w i t h  of  p  [{-)  confounding p^(to{-)  I f an a g e n t ' s p r e f e r e n c e s a r e markedly  \  i n s t a t e s tn and c o , t h e n u m e r i c a l v a l u e o f t h e excess  demand w i l l  1  2  be v e r y v o l a t i l e and d i s c o n t i n u o u s a t e = \ as p  s u d d e n l y becomes i n v e r t i b l e . 6 = j|  if  u  Q  This information d i s c o n t i n u i t y a t  i s n o t removable and g e n e r a l l y l e a d s t o a jump d i s c o n -  t i n u i t y o f demand a t 0 = |  .  These d i s c o n t i n u i t i e s  tend to  foil  the a p p l i c a t i o n o f f i x e d p o i n t theorems t o e s t a b l i s h t h e e x i s t e n c e of  equilibrium. In a d d i t i o n t o t h e s e c o n t i n u i t y  problems, there are other  problems (which may be termed " m e a s u r a b i l i t y p r o b l e m s " ) e s t a b l i s h i n g the e x i s t e n c e o f a r a t i o n a l  expectations  with  equilibrium.  We have a l r e a d y argued t h a t agent i ' s e x c e s s demands w i l l be measurable w i t h r e s p e c t t o h i s i n f o r m a t i o n A^VP"^(A); t h a t i s , h i s demands w i l l ally receives.  vary only according to the information On t h e o t h e r h a n d , d i f f e r e n t  pieces of  he a c t u information  17.  will  g e n e r a l l y l e a d to d i f f e r e n t  demand f u n c t i o n w i l l  demands, so t h a t h i s excess  o f t e n g e n e r a t e the whole a - a l g e b r a >A.jVP~^ ( A ) .  For a more c o n c r e t e i n t e r p r e t a t i o n , space ^ i s f i n i t e ,  suppose the  so t h a t we need o n l y c o n s i d e r the  t h a t g e n e r a t e the i n f o r m a t i o n a - a l g e b r a s .  (A f i n e r  c o r r e s p o n d s to s u p e r i o r i n f o r m a t i o n , e t c . ) o f n o t a t i o n , l e t A^ be agent i ' s  members o f h i s p a r t i t i o n .  partitions partition  With an o b v i o u s change  exogenous i n f o r m a t i o n  so t h a t he can d i s t i n g u i s h s t a t e s toeft  by p i . e . ,  state  only i f  they a r e i n  L e t p~^(A) be the p a r t i t i o n  the p a r t i t i o n  common r e f i n e m e n t o f A^ and P~^(A).  on d i f f e r e n t  the whole  partition).  generated  Then A.VP'^A) i s  To say t h a t agent i ' s  demand f u n c t i o n i s AjVP ^ ( A ) - m e a s u r a b l e means t h a t i t on members o f the p a r t i t i o n .  different  c o n s i s t i n g of equivalence sets of states  co which map to the same p r i c e under P.  different  partition,  the excess  i s constant  In g e n e r a l , excess demands w i l l  members o f the p a r t i t i o n  (i.e., will  be  generate  The e q u i l i b r i u m c o n d i t i o n i s t h a t e x c e s s demands sum t o z e r o , for a l l  coeft .  But e x c e s s demands t h a t v a r y on d i f f e r e n t  (or a - a l g e b r a s ) w i l l  not,  i n g e n e r a l , sum t o a c o n s t a n t  partitions function.  For e x a m p l e , i n a two-agent economy, we cannot have an e q u i l i b r i u m where one agent v a r i e s h i s excess demands on a f i n e r different)  partition  (after  simply,  than the o t h e r , because such excess demand  cannot sum t o z e r o f o r a l l information  (or  coefi..  If  we can r u l e o u t  differential  t a k i n g i n t o a c c o u n t endogenous p r i c e  then we may g e t an e q u i l i b r i u m .  information)  That i s , , we r e q u i r e , of.; an^e.quit-14.brium  p that  vp~'(A)..= AjVp~'(A) f o r a l l i , j . Hence, we have  A ^ p ' ^ A ) c A C (A V " (A))V...V(A VP" (A))= A.VP (A), 1  1  1  p  _1  I  so that A = A^vp (A) f o r a l l i . That i s i n e q u i l i b r i u m , a l l -1  agents are f u l l y informed.  I f the A., d i f f e r s u f f i c i e n t l y  t h i s a l s o tends to imply that P""'(A) =A,  that i s , a l l  information i s revealed by p r i c e s .  If a rational  e q u i l i b r i u m e x i s t s with  we s h a l l say that a f u ! l y  P  _ 1  (A) = A,  informing r a t i o n a l expectations e q u i l i b r i u m (FRE)  expectations  exits.  This measurability argument suggests t h a t , i f agents " l i k e to use a l o t of information" i n formulating t h e i r demands ( i . e . , S(p(*)>*> A..vp  -1  (A))  generates the whole c-algebra A_.'vp (A))., _1  agents must be f u l l y informed f o r e q u i l i b r i u m to e x i s t , and i f the A 's vary s i g n i f i c a n t l y across agents, the only way to provide -1 3) a l l information i s to have P ( A ) = A , a FRE. ;  I f under perturbations o f to, a FRE continues to e x i s t , there are no information d i s c o n t i n u i t i e s so that the c o n t i n u i t y and measurability problems are solved simultaneously with FRE's. In the f o l l o w i n g chapters we study the existence of FRE's only. There i s a straightforward procedure f o r v e r i f y i n g the existence of a FRE. demand functions  One simply solves f o r the f u l l y - i n f o r m e d  5.j(p,<",A).  The market c l e a r i n g condition then  defines P, from which i t i s then, i n p r i n c i p l e , possible to check whether  A  =. p " ^ ( A ) .  i n p r a c t i c e , there may be simple parameters  which are s u f f i c i e n t f o r defi n i n g c o n d i t i o n a l p r o b a b i l i t i e s or summarizing the source of uncertainty, such as s u f f i c i e n t s t a t i s t i c s  19.  drawn from i n d e p e n d e n t , i d e n t i c a l l y A sufficient an i n v e r t i b l e  f u n c t i o n o f these p a r a m e t e r s .  T h i s i s the  The term " r a t i o n a l  expectations  e x p e c t a t i o n s " was c o i n e d i n a paper by  H i s model was the f i r s t o f a s e r i e s o f macroeconomic  models d e v e l o p e d by v a r i o u s a u t h o r s . homogeneous i n f o r m a t i o n .(A.= To enhance the e m p i r i c a l  These models g e n e r a l l y have  A, V i ) and homogeneous b e l i e f s  testability  of these models, i t  assumed t h a t P a l s o r e p r e s e n t s N a t u r e ' s " t r u e " so t h a t ex p o s t p r o b a b i l i t y  s t r o n g e r c o n c l u s i o n s i n macro m o d e l s . see S h i l l e r [ 1 9 7 8 ] ,  and i n f o r m a t i o n  is  laws a r e the same as ex a n t e  i s b a s i c a l l y a strong consistency c r i t e r i o n  (P..= P , \ / i ) .  probability  In t h e s e m o d e l s , the assumption o f r a t i o n a l  literature,  technique  chapters.  Other papers c o n c e r n i n g r a t i o n a l  laws.  random v a r i a b l e s .  c o n d i t i o n f o r the e x i s t e n c e o f a FRE i s t h a t p be  used i n the f o l l o w i n g  Muth [ 1 9 6 1 ] .  distributed  law. probability  expectations  t h a t i s used to  For a good r e v i e w o f  Ross [1978] i n models o f the term s t r u c t u r e o f i n t e r e s t  rates.  r a t e s , they used the assumption o f homogeneous and c o r r e c t  interest beliefs  r a t e p r o c e s s t h a t agents assume h o l d when  demands to the p r o c e s s t h a t a c t u a l l y r e s u l t s ,  a g e n t s ' demands.  this  and  In a g e n e r a l e q u i l i b r i u m model o f the s t o c h a s t i c p r o c e s s o f  formulating  obtain  In f i n a n c e , the homogeneous b e l i e f s  assumption has been used by C o x , I n g e r s o l l  to r e l a t e the i n t e r e s t  usually  given  The m i c r o - e c o n o m i c and g e n e r a l e q u i l i b r i u m a p p l i c a t i o n s rational tial  e x p e c t a t i o n s assumptions g e n e r a l l y c e n t e r around d i f f e r e n -  i n f o r m a t i o n and homogeneous (and c o r r e c t )  beliefs.  Here  the c e n t r a l q u e s t i o n i s o f t e n the e x i s t e n c e o f e q u i l i b r i u m . An example o f such a d i f f e r e n t i a l  In t h i s m a r k e t , the s e l l e r has b e t t e r  the q u a l i t y o f h i s c a r than has the b u y e r . method o f s i g n a l l i n g q u a l i t y ,  expec-  markets by A k e r l o f information  about  In the absence o f some  the owner o f a good c a r  r e c e i v e a p r i c e c o r r e s p o n d i n g to an average ( i . e . , He withdraws h i s c a r  -  information rational  t a t i o n s model i s the "lemons" model o f used c a r [1970].  of  will  only  inferior)  car.  from the m a r k e t , l e a v i n g a l o w e r average  q u a l i t y , and,' i n a l i k e manner, the owners o f the n e x t grade o f c a r s l e a v e the m a r k e t , f i n a l l y on the m a r k e t .  l e a v i n g o n l y the l o w e s t q u a l i t y  In t h i s c a s e , i n f o r m a t i o n  t h a t a c a r - i s o f f e r e d on the market ( f o r  "lemons"  i s conveyed by the a given p r i c e ) .  fact  A t any  g i v e n p r i c e , t h e r e i s p o s i t i v e s u p p l y and p o s i t i v e demand, but two a r e never e q u a l , so the market  the  fails.  In f i n a n c e , t h e r e a r e models o f c a p i t a l  s t r u c t u r e by L e l a n d - P y l e  [1977] and Ross [1977, 1 9 7 8 a ] - a n d - o f c r e d i t r a t i o n i n g by J a f f e e - R u s s e l l [1976] and H e i n k e l [ 1 9 7 8 ] .  These models use d i f f e r e n t i a l  information  t o e x p l a i n v a r i o u s e m p i r i c a l l y observed phenomena t h a t h i t h e r t o c o u l d o n l y be e x p l a i n e d by i n s t i t u t i o n a l costs.  rigidities  The c a p i t a l s t r u c t u r e models show how i t  f i r m to have an o p t i m a l c a p i t a l s t r u c t u r e  like  transactions  is possible for a  in a tax-free world,  in  c o n t r a d i c t i o n to the M o d i g l i a n i - M i l l e r i r r e l e v a n c e theorem.  In  the L e l a n d - P y l e m o d e l , the p r o p o r t i o n o f e q u i t y f i n a n c e d by i n s i d e r s s i g n a l s i n f o r m a t i o n about the random r e t u r n s on c a p i t a l  investment.  This r e s u l t s i n a f u l l y  equilibrium  informing r a t i o n a l  expectations  i n which e n t r e p r e n e u r s o v e r - i n v e s t i n t h e i r f i r m r e l a t i v e to what t h e i r optimal  i n v e s t m e n t would be i f a l l  the same i n f o r m a t i o n .  agents exogeneously have  This occurs because, with d i f f e r e n t i a l  in-  f o r m a t i o n the b e n e f i t s o f i n s i d e i n v e s t m e n t i n the f i r m a r e not o n l y the f u t u r e  " r e t u r n s , but a l s o the c u r r e n t r e t u r n s o f s e l l i n g  o f the f i r m a t the h i g h e r p r i c e t h a t r e s u l t s when i t as a h i g h - r e t u r n  firm.  This d i s t o r t i o n  i s due to the  part  is classified differential  i n f o r m a t i o n , not to a l a c k o f p r i c e - t a k i n g b e h a v i o u r . In the Ross m o d e l s , managers have s u p e r i o r i n f o r m a t i o n  about  f i r m type and have compensation based i n c e n t i v e schemes t h a t a l l o w the market to p e r c e i v e f i r m type by o b s e r v i n g the manager's compensation formula.  In s e v e r a l o f the Ross m o d e l s , the compensation  scheme i s based on the c u r r e n t v a l u e o f the f i r m , as w e l l as the f u t u r e v a l u e o f the e q u i t y . managers t a i l o r  By v a r y i n g the f i r m ' s debt  level,  t h e i r compensation scheme to s i g n a l f i r m t y p e ,  y i e l d i n g an o p t i m a l debt  level.  The H e i n k e l model uses a b o r r o w e r ' s c h o i c e o f the amount o f debt to s i g n a l h i s p r e v i o u s l y p r i v a t e i n f o r m a t i o n about how r i s k y his project i s .  R i s k y borrowers would l i k e to borrow a l a r g e amount,  since, with limited l i a b i l i t y  they can b e n e f i t s i g n i f i c a n t l y by h i g h  p r o j e c t r e t u r n s , but a r e not p e n a l i z e d s i g n i f i c a n t l y f o r low  returns.  22. By r a t i o n i n g borrowers the l i m i t e d l i a b i l i t y and l e n d e r s w i l l  (i.e.,  keeping the f a c e v a l u e o f debt  feature i s of l e s s value of r i s k y  o n l y l e n d to low r i s k b o r r o w e r s .  borrowers r e v e a l t h e i r tinct interest/loan  t r u e type ( " s e l f - s e l e c t " )  size pairs.  low),  borrowers,  Thus, the by c h o o s i n g d i s -  However, l o w - r i s k borrowers  r a t i o n e d to a s m a l l e r l o a n s i z e than they would choose i f c l a s s e s were exogenously r e v e a l e d .  are  risk  The J a f f e e - R u s s e l l paper has  some s i m i l a r n o t i o n s , but emphasizes " h o n e s t " v e r s u s  "dishonest"  borrowers. These models o f debt markets and o f c a p i t a l  structure,  as  w e l l as r e l a t e d models o f i n s u r a n c e markets by R o t h s c h i l d - S t i g l i t z [1976] have the f e a t u r e  t h a t , a l t h o u g h agents a r e p r i c e - t a k e r s ,  informed a g e n t s ' a c t i o n s a f f e c t goods they a r e o f f e r i n g  the " m a r k e t ' s " p e r c e p t i o n s o f  ( e . g . , " f i r m t y p e " ) , and hence the  litera-  t u r e c e n t e r s on i n c e n t i v e schemes, moral h a z a r d , agency and mation s i g n a l l i n g  infor-  behaviour.  There a r e o t h e r m o d e l s , however, i n which agents do not i n ways to e x p l i c i t l y  act  encourage o r d i s c o u r a g e the d i s s e m i n a t i o n  their private information  to the m a r k e t .  [1976] models a s i t u a t i o n  i n which agents have d i f f e r e n t  p r i c e i s determined by the aggregate c o n t e n t o f t h a t (which i s s u p e r i o r to each a g e n t ' s p r i v a t e the r e l e v a n t a g g r e g a t e i n f o r m a t i o n  of  For e x a m p l e , Grossman  t i o n about the n e x t - p e r i o d v a l u e o f a r i s k y s e c u r i t y .  price alone.  the  informa-  The c u r r e n t information  information).  Moreover,  can be i n f e r r e d by o b s e r v i n g  T h i s l e a d s to one d i s t u r b i n g  f e a t u r e o f Grossman's m o d e l ,  23.  as p o i n t e d o u t by Grossman [1976, p . 5 8 2 ] , Stiglitz  [1976].  If  Kraus [1976] and Grossman-  the p r i v a t e c o l l e c t i o n o f i n f o r m a t i o n  but the o b s e r v a t i o n o f p r i c e s i s c o s t l e s s , agents w i l l private  information,  expecting that i t w i l l  which, of course, i t w i l l  not,  if  not  nobody c o l l e c t s  Rothschild [1976],  i n which e q u i l i b r i u m f a i l s For s i m p l i c i t y ,  There a r e  a n d . J o r d a n - R a d n e r [1977]  to o b t a i n i n r a t i o n a l  expectations models. agents:  In a f i n i t e s t a t e - s p a c e m o d e l , the  agents can d i s t i n g u i s h s t a t e s i n a more r e f i n e d p a r t i t i o n uninformed a g e n t s .  However, p r o b a b i l i t y  informed  than  the  b e l i e f s a r e homogeneous i n  the sense t h a t both agents agree on the p r o b a b i l i t i e s the c o a r s e r , uninformed  collect  information.  these models deal w i t h two ( c l a s s e s o f )  i n f o r m e d and u n i n f o r m e d .  costly,  be r e v e a l e d i n p r i c e s ,  At l e a s t Grossman's model y i e l d s an e q u i l i b r i u m . models by Green [ 1 9 7 7 ] ,  is  o f events  in  partition.  In t h e s e m o d e l s , the uninformed agents e i t h e r become informed o r n o t when they o b s e r v e e q u i l i b r i u m p r i c e s .  If  t h e i r demands a r e such t h a t p r i c e s a r e not f u l l y they a r e not i n f o r m e d , o n l y f u l l y  they a r e  informed,  i n f o r m i n g , but  i n f o r m i n g p r i c e s can  equilibrate  the economy, s i n c e informed a g e n t s ' e x c e s s demands are d i f f e r e n t each member i n the r e f i n e d p a r t i t i o n . f o r c e the n o n - e x i s t e n c e o f a r a t i o n a l  if  Thus, m e a s u r a b i l i t y  problems  expectations equilibrium.  the J o r d a n - R a d n e r m o d e l , n o n - e x i s t e n c e o c c u r s even under  in  In  perturbations  o f the u n d e r l y i n g parameters o f the economy — t h a t i s , - n o n - e x i s t e n c e i s , in a sense,  generic.  On the o t h e r hand, t h e r e are g e n e r i c e x i s t e n c e r e s u l t s  for  F R E ' s by Radner [1978] and A l l e n [1978, 1 9 7 9 ] , which a r e based on theorems i n d i f f e r e n t i a l  topology.  The g e n e r a l r e s u l t i s  the d i m e n s i o n o f the space o f p r i c e s must ( s u b s t a n t i a l l y ) the dimension o f the i n f o r m a t i o n restrictions  exceed  These d i m e n s i o n a l i t y  a r e v i o l a t e d i n the J o r d a n - R a d n e r c o u n t e r e x a m p l e .  These r e s u l t s have a l l information  space.  that  only.  a l s o important  addressed the communication o f  probability  I n f o r m a t i o n about p r e f e r e n c e s and endowments  is  and the q u e s t i o n o f whether o r not t h i s can be c o n -  veyed by p r i c e s i s d i s c u s s e d i n K r a u s - S i c k [ 1 9 7 9 a , 1979b, 1 9 8 0 ] . The f o l l o w i n g  c h a p t e r s d i s c u s s those r e s u l t s as w e l l  o f those r e s u l t s .  as e x t e n s i o n s  .25.  F o o t n o t e s to C h a p t e r 2  1.  A rigourous understanding of measure-theoretic p r o b a b i l i t y  theory  i s n o t r e q u i r e d to u n d e r s t a n d the t h r u s t o f the argument p r e s e n t e d here. II,  The t h e o r y i s p r e s e n t e d ' i n , f o r example, F e l l e r [1966, V o l .  C h a p t e r s IV and V] and r e f e r e n c e s t h e r e i n .  s e l e c t i o n o f a s t a t e u>eo, the s y s t e m .  Roughly s p e a k i n g ,  c o r r e s p o n d s t o a random r e a l i z a t i o n o f  The o - a l g e b r a 8 i s a f a m i l y o f s u b s e t s ( c a l l e d e v e n t s )  o f o., which s a t i s f i e s s p e c i a l c o n d i t i o n s .  The p r o b a b i l i t y measure  P.,- : B -> [ 0 , 1 ] a s s i g n s to each event B e B a n o n - n e g a t i v e o f i t s o c c u r r e n c e , with-P-Cn) = 1.  The p r o b a b i l i t y  o f n t h a t i s n o t an e v e n t i s u n d e f i n e d . f u n c t i o n n : o, -> &  o f any s u b s e t  A random v a r i a b l e i s a  ( f o r some n) which i s B - m e a s u r a b l e .  n  probability  That i s ,  f o r any ( a - ! , . . . , a ) & ' n  n  e  {co e ft | n . (co) < _ a . , j = l , . . . , h } e B , J  J  J.L.  where n.(oo) i s t h e j 3  component o f (u>). n  Roughly s p e a k i n g , t h i s  means t h a t n does n o t v a r y too much - - any change i n t h e v a l u e o f rt can be c a p t u r e d by an event i n B . 2.  F o r example, l e t n : &  &  n  be a random v a r i a b l e .  example, F e l l e r [1966, V o l . I I ,  pp. 1 6 0 - 1 6 2 ] ) ,  Then ( s e e , f o r  the c o n d i t i o n a l  e x p e c t a t i o n E.(nU^) i s t h e A.j-measurable random v a r i a b l e 5 : n -»- &  n  :such t h a t / E . Q ^ ^ J A e C w O d P i U )  =  J A  n(c )dP ( ) J  i  u  = E.(l n) A  for a l l  A A . where 1„ : o, -»- &. i s t h e s e t i n d i c a t o r f u n c t i o n d e f i n e d by e  1 i f co e A 0 i f to f. A  Note t h a t , since F..,-(ij | Aj) i s a random v a r i a b l e i t depends on the r e a l i z a t i o n coeft (n| Aj  and should r e a l l y be w r i t t e n as, say  . Following conventional usage, we suppress the  e x p l i c i t references to CJ.  I f p r o b a b i l i t y d e n s i t i e s e x i s t , or  p r o b a b i l i t y i s d i s c r e t e , t h i s i s i n accordance with the usual notion of c o n d i t i o n a l expectation or p r o b a b i l i t y .  Also, i f  AeB i s an event, define  (n| ) i s  = {<t>,A,ft\A,ft}.  Since  A.-measurable, i t i s constant on the event A and the constant i s unique i f A has p o s i t i v e p r o b a b i l i t y . constant value on A.  Let E.(n|A) be that  This conforms with our conventional notion  of the non-random c o n d i t i o n a l expectation given an event of p o s i t i v e probability. To c o n d i t i o n on a random v a r i a b l e , p say, i s to c o n d i t i o n on the o-algebra generated by p,  namely that a-algebra  generated  by the events {weft| p.'(co) < a .%j = 1,.. .n} :  for a r b i t r a r y (a-j,... ,a ) § n  &. n  I ffti s f i n i t e , the a-algebra  generated by n i s that which i s generated by the p a r t i t i o n o f ft generated by inverse images of points i n £ 1 S  n  under p.  The p a r t i t i o n  {{co : p(co) = p} : .pe& } n  The a-algebra generated by p d i s t i n g u i s h e s only those points which are mapped to d i f f e r e n t points by p, and hence i s the information conveyed by observing p.  T h i s i s not to s u g g e s t t h a t n o n - f u l l y  informing  e x p e c t a t i o n s e q u i l i b r i a do not e x i s t . dimensions o f u n c e r t a i n t y  If  t h e r e a r e two  (a =•£. ) but an informed  demand and hence p r i c e conveys o n l y a u n i v a r i a t e o f the u n c e r t a i n t y ,  it  rational  agent's function  i s p o s s i b l e f o r e q u i l i b r i u m to  exist.  T h i s o c c u r s , f o r example, i n Grossman [ 1 9 7 7 ] , i n which r a n dom s u p p l y i s added to u n c e r t a i n t y risky asset.  about the v a l u e o f a  The informed a g e n t s ' demands are u n i v a r i a t e  in  n a t u r e and generate a " o n e - d i m e n s i o n a l a - a l g e b r a " ( i . e . , a a - a l g e b r a g e n e r a t e d by o n e - d i m e n s i o n a l  translations  of a  m a n i f o l d , so t h a t t h e r e i s o n l y one r e m a i n i n g degree o f freedom on which v a r i a t i o n can o c c u r i n a ). optimal tional  In h i s m o d e l ,  demands f o r the uninformed agent are merely c o n d i expectations of optimal  informal  demands, and hence  can g e n e r a t e the same one d i m e n s i o n a l © - a l g e b r a , excess demands to sum to z e r o .  If  allowing  uninformed agents have  d i f f e r e n t p r e f e r e n c e s , than informed a g e n t s , o r have some information  t h a t " i n f o r m e d " agents do not h a v e , then u n i n -  formed demands may n o t generate the same a - a l g e b r a as a g e n t s , o r have some i n f o r m a t i o n  informed  t h a t " i n f o r m e d " agents do  n o t h a v e , then uninformed demands may n o t generate the same a - a l g e b r a as informed demands and the m e a s u r a b i l i t y  problem  p r e v e n t s the e x i s t e n c e o f e q u i l i b r i u m .  interesting  to study the r o b u s t n e s s o f t h i s expectations  equlibrium.  I t would be  non-fully  informing  rational  Chapter 3  R e v e l a t i o n o f Aggregate P r e f e r e n c e Parameters  Introduction In t h i s c h a p t e r , we study the q u e s t i o n o f w h e t h e r , i n a m u l t i p e r i o d economy, c u r r e n t p r i c e s r e v e a l enough i n f o r m a t i o n about the aggregate p r e f e r e n c e parameters o f the economy to r e s o l v e some o f the u n c e r t a i n t y about f u t u r e p r i c e s . expectations context:  T h i s i s done i n a r a t i o n a l  the demands l e a d i n g to the f o r m a t i o n  of  p r i c e s must be c o n s i s t e n t w i t h the i n f o r m a t i o n conveyed by p r i c e s . M o t i v a t i o n f o r t h i s work comes from three, d i r e c t i o n s . One l i n e o f m o t i v a t i o n comes from the Arrow [1953] paper on the o p t i m a l  a l l o c a t i o n of r i s k - b e a r i n g .  Arrow f i r s t showed t h a t  i n an economy w i t h C commodities and S s t a t e s o f the w o r l d , SC state-contingent  ("Arrow-Debreu")  state-commodity p a i r w i l l  s e c u r i t i e s c o r r e s p o n d i n g to each  achieve a Pareto e f f i c i e n t  equilibrium.  He then e s t a b l i s h e d t h a t the same a l l o c a t i o n can be o b t a i n e d by S+C markets:  S markets f o r s t a t e - c o n t i n g e n t w e a l t h and,.'once  a p a r t i c u l a r s t a t e i s r e v e a l e d , C markets f o r the commodities^ An o b j e c t i o n , v o i c e d , f o r e x a m p l e , by Dreze [1970, p. 144] and Nagatani [ 1 9 7 5 ] , will  i s t h a t t h i s r e d u c t i o n i n the number o f markets  r e s u l t i n the same a l l o c a t i o n o n l y i f  prices w i l l  prevail  agents know what commodity  i n each s t a t e o f n a t u r e , b e f o r e i t  i s revealed  and hence b e f o r e the market opens to generate these p r i c e s .  If  c u r r e n t p r i c e s can r e v e a l p r e f e r e n c e p a r a m e t e r s , they may r e v e a l these s t a t e - c o n d i t i o n a l  f u t u r e p r i c e s , a l l o w i n g an o p t i m a l  allocation  w i t h a reduced number o f markets i n the manner e n v i s i o n e d by A r r o w .  A n o t h e r l i n e o f m o t i v a t i o n i s p r o v i d e d by Grossman [ 1 9 7 6 ] , i n which agents come to a market w i t h d i v e r s e i n f o r m a t i o n  about  a r i s k y a s s e t and the market c l e a r i n g p r o c e s s a g g r e g a t e s t h a t i n f o r m a t i o n and conveys the aggregate parameter, v a l u e , v i a p r i c e signals.  In t h i s c h a p t e r , agents come to market w i t h d i v e r s e  p r e f e r e n c e s and the market p r i c e conveys an aggregate p r e f e r e n c e v a l u e r a t h e r than aggregate p r o b a b i l i t y  i n f o r m a t i o n , which i s  i m p o r t a n t i n p r e d i c t i n g next p e r i o d p r i c e s . A t h i r d l i n e o f m o t i v a t i o n comes from the r e c e n t l i t e r a t u r e the r e c o v e r a b i l i t y o f u t i l i t y f u n c t i o n s - - s e e , e . g . , Dybvig  on  [1979],  Dybvig and P o l e m a r c h a k i s [1979] and G r e e n , Lau and P o l e m a r c h a k i s [ 1 9 7 9 ] . The g e n e r a l q u e s t i o n a d d r e s s e d i n t h i s l i t e r a t u r e  i s : ' "Given c e r t a i n  r e s t r i c t i o n s on the a d m i s s i b l e c l a s s o f u t i l i t y f u n c t i o n s , and an opportunity  to o b s e r v e an a g e n t ' s demand f u n c t i o n on a c e r t a i n s e t  o f p r i c e s and incomes, when can the u t i l i t y f u n c t i o n be determined ('recovered')?"  F o r e x a m p l e , Dybvig s t u d i e s the c l a s s o f  additive  u t i l i t y f u n c t i o n s f o r two goods ( e . g . , complete markets and von Neumann-Morgenstern u t i l i t y ) and f i n d s c o n d i t i o n s under which the  utility  f u n c t i o n i s r e c o v e r a b l e g i v e n two Engel c u r v e s ( i m p o s s i b l e ) o r g i v e n t h r e e Engel c u r v e s ( g e n e r i c a l l y p o s s i b l e ) o r g i v e n demands a l o n g two r a y s through the o r i g i n  (always p o s s i b l e , p r o v i d i n g the s l o p e  o f the u t i l i t y f u n c t i o n i s bounded as consumption goes t o z e r o ) . Dybvig has analogous r e s u l t s f o r u t i l i t y f u n c t i o n s w i t h more than two goods ( o r s t a t e s ) .  Dybvig a l s o s t u d i e s the r e c o v e r a b i l i t y  von Neumann-Morgenstern u t i l i t y from demands f o r l o t t e r i e s s i n g l e r i s k y a s s e t and f i n d s t h a t the main c o n d i t i o n f o r  of  over a recoverability  i s t h a t m a r g i n a l u t i l i t y be bounded as consumption goes to z e r o .  There are two important c h a r a c t e r i s t i c s of these r e c o v e r a b i l i t y results,, i n contrast to the present research: 1.  U t i l i t y functions are recovered from demand f u n c t i o n s , which are  not observable from p r i c e data i n a general e q u i l i b r i u m s e t t i n g (unless m u l t i p l e observations are a v a i l a b l e f o r econometric estimation). 2.  They require an i n f i n i t e number of observations to s p e c i f y whole  Engel curves, f o r example, unless the Engel curves are previously known to have a s p e c i f i c f u n c t i o n a l form, i n which case only enough observations to evaluate a l l parameters are required. we use a general e q u i l i b r i u m s e t t i n g and recover observing one r e a l i z a t i o n of the economy.  In c o n t r a s t ,  information by  This i s obtained at the  expense, of course, of having to previously s p e c i f y the f u n c t i o n a l form of the u t i l i t y f u n c t i o n . The f i r s t models i n t h i s chapter are i n a two period s t a t e p r e f e r ence s e t t i n g a r i s i n g from complete markets.  The l a s t model of the  chapter has two periods and incomplete markets with normally d i s t r i b u t e d s e c u r i t y returns. Notation We deal with a pure exchange economy.and a s i n g l e good ("money"). There are I agents ( i = 1,  I ) , and three dates (0,1,2).  Securities  which are the only traded o b j e c t s , pay o f f i n money i n C f i n a l (date 2) states (c = 1, ..., C) or i n S intermediate (date 1) states (s = 1 There are homogeneous p r o b a b i l i t y b e l i e f s at date 0 about the j o i n t d i s t r i b u t i o n of intermediate and f i n a l s t a t e s . ir  s  7T  ii  S C  c  These b e l i e f s are:  =  Pr (intermediate s t a t e s)  =  Pr ( f i n a l state c|intermediate s t a t e s)  E  Pr ( f i n a l state c) = £ TT i r „ s s sc  One may think o f the intermediate states  s  as presenting new proba-  b i l i t y information causing each agent to update h i s p r o b a b i l i t y estimates about f i n a l state c from T T to i r c  sc  .  An agent i s endowed with money a t date 0 and makes sequential p o r t f o l i o decisions a t date 0 and date 1 f o r , r e s p e c t i v e l y , an i n i t i a l p o r t f o l i o of claims on money i n intermediate states and a f i n a l p o r t f o l i o of claims on money i n f i n a l states. y.j y. 1 S  y.  These are denoted by:  = endowed wealth of agent i • = payoff to agent i ' s i n i t i a l p o r t f o l i o i f i n t e r mediate s t a t e s p r e v a i l s a t date 1 = payoff to agent i ' s f i n a l p o r t f o l i o i f intermediate state s p r e v a i l s a t date 1 (when f i n a l p o r t f o l i o is purchased) and f i n a l s t a t e c p r e v a i l s at date 2. The aggregate supply of these claims i s X = Z|X - . Without loss o f g e n e r a l i t y , we assume X is independent of s. 1  se  Markets c o n s i s t o f two types o f s e c u r i t i e s : Type " f " i s a r i s k l e s s bond maturing a t the next date with a y i e l d of zero ( i . e . , cash), with zero net supply.  Type  "m" i s a proportional share o f the s o c i a l endowment a t the next date ( i . e . , the market p o r t f o l i o ) , with unit net supply. Define prices o f the market p o r t f o l i o by: PQ  =  date 0 price o f market p o r t f o l i o i n terms o f cash numeraire.  P  s  =  date 1 price o f market p o r t f o l i o i n state terms o f a s t a t e  Let  m.  =  individual  s  s, i n  cash numeraire.  i ' s endowed f r a c t i o n a l holding o f the  market p o r t f o l i o a t date 0 (£.m. = 1 ) ,  f  = i n d i v i d u a l i ' s d o l l a r holding of cash i n the  i 0  p o r t f o l i o selected a t date 0, m^Q = i n d i v i d u a l i ' s f r a c t i o n a l holding of next date s o c i a l endowment i n the p o r t f o l i o selected at date 0, f  is' is m  s i m i l a r l y f o r the f i n a l p o r t f o l i o selected at date  E  1 i n intermediate state s. Then intermediate and f i n a l payoffs are, r e s p e c t i v e l y , ( )  *is= i0  3 J  <-> 3  2  f  x  isc  =  f  +  is  +  m  m  iO s p  is c • X  Budget constraints f o r dates 0 and 1 are, r e s p e c t i v e l y , +p m  (3.3)  f  i Q  (3.4)  f  i s +  0  p m s  = p m.  iQ  0  = y.  is  .  s  Market c l e a r i n g r e l a t i o n s are: (3.5)  Z.f.  (3.6)  ^  - Z,f  Q  = i  0  Z  -  H  i is m  = 1  0 -  There i s no consumption before the f i n a l period (and hence no natural inter-temporal discount r a t e ) , so t h a t , i n market M, there i s a need f o r S numeraires f o r date 1 and one numeraire f o r date 0. In (3.3) and (3.4), these numeraires have been chosen to be the "cash" p o r t f o l i o relevant f o r that date and s t a t e .  For s i m p l i c i t y  we may define y^ E p^m^ so that (3.3) becomes: (3.7)  f  i Q  +p m 0  iQ  =  y i  .  We may think of t h i s market s e t t i n g as a r i s i n g  at each date  from complete markets of s e c u r i t i e s paying o f f at the next date.  The complete market s t r u c t u r e s e c u r i t i e s because a l l  agents ( w i l l ) have l i n e a r r i s k  (LRT) w i t h the same s l o p e . Morgenstern  degenerates to a market f o r  That i s , i f  two  tolerance  an agent has von Neumann-  u t i l i t y U(x) f o r w e a l t h x , then h i s r i s k  tolerance  ( a t x) i s the i n v e r s e o f h i s a b s o l u t e r i s k a v e r s i o n : 1  -  =  R (x)  U'(x)  "  A  U"(x)  We s h a l l assume t h a t agents have l i n e a r r i s k t o l e r a n c e w i t h the same s l o p e c o e f f i c i e n t (3.8)  ("cautiousness"),  i.e.:  U.'(x) " U^T*)  =  a  i  +  X x  T h i s i s n e c e s s a r y and s u f f i c i e n t and s u r r o g a t e f u n c t i o n s aggregation ( c f .  (cf.  for l i n e a r sharing rules  W i l s o n [1968]) and hence r e s u l t s  in  R u b i n s t e i n [1974] and Brennan-Kraus [1978]) as  w e l l as two-fund monetary s e p a r a t i o n ( c f .  C a s s - S t i g l i t z [1970]).  With two-fund monetary s e p a r a t i o n , t r a d i n g i n a complete market will  degenerate i n t o t r a d i n g  and the market p o r t f o l i o loss of g e n e r a l i t y ,  i n two p o r t f o l i o s :  of a l l  risky assets.  cash  We may assume, w i t h o u t  t h a t cash has a net zero s u p p l y , because the  n e t . s u p p l y o f cash can be absorbed i n t o the market merely s h i f t i n g  its  ("money")  return distribution  portfolio,  by a c o n s t a n t .  A power u t i l i t y economy Suppose a l l exponent)  agents have extended power u t i l i t y  f o r consumption a t date 2 .  ( w i t h the same  S p e c i f i c a l l y , the  utility  f o r agent i f o r consuming x a t date 2 i s : (3.9)  U.(x)  = Y"  1  (e +x) i  Y  i = 1  I  where 0 f y < 1.  This i s a concave, increasing u t i l i t y function  d i s p l a y i n g decreasing absolute r i s k a v e r s i o n . ^ state s agent i chooses a p o r t f o l i o (f - » n  max  (e.+f. +m. X j '  C SC  < is' is f  m  to solve:  s  Y  IS c  IS  1  v  At date 1 i n  }  subject t o : (3.4)  f  i  s  + p  m  s  =y.  is  .  s  Note that ^e have used (3.2) to s u b s t i t u t e x the o b j e c t i v e f u n c t i o n . S  (3.10)  p  = f^ + m X s  is  c  into  The f i r s t order conditions y i e l d  c sc c n  i s c  X  0 + y + m . ( x - p ) Jh - i i  e  s  is  i yis +  + m  c  s  ¥-1  is( c-P ) X  S  This i s a nonlinear equation i m p l i c i t l y g i v i n g no closed form s o l u t i o n for the demand m^  as a function o f p  s  $  to v e r i f y that (3.10) holds f o r a l l i (3.11)  ( 3  '  Q  m-„ is  (3.13)  =  0  2) .  i = 1,  + y  c  s c  xfe.+xj^ c  A  Vsc A  i f and only i f  i is „0 •, - •  i n S  However, r , ii t i s easy  I and  A + P s  1 2 )  P  and y ^ .  = E  I  0  where  ( A c) 0  i  - 1  c  + X  Y  .  Summing (3.1) over i and noting (3.5) and (3.6) y i e l d s ^ ^ y  = i s  P  s  Hence, summing (3.11) over i y i e l d s E . rn. = 1, so that the market f o r s  the r i s k y asset clears a t date 1. for  By Walras' law, the market  the r i s k l e s s asset also c l e a r s , so (3.11) and (3.12) represents  an e q u i l i b r i u m , where f . i s computed from (3.4). g  Note that (3.11)  does not g i v e a demand f u n c t i o n f o r the r i s k y a s s e t , s i n c e s o l v e s ( 3 . 1 0 ) o n l y when (3.12) (3.12)  does h o l d and ( 3 . 1 1 )  holds.  However, i n  it  equilibrium,  g i v e s the c o r r e c t s h a r i n g r u l e o r  numerical v a l u e o f demand. A n o t h e r way o f o b t a i n i n g ( 3 . 1 2 )  i s to compute i t  as the  m a r g i n a l r a t e o f s u b s t i t u t i o n . , between the r i s k y and the r i s k l e s s a s s e t f o r a r e p r e s e n t a t i v e o r aggregate i n v e s t o r A w i t h a l l w e a l t h ^so t h a t m  = 1 and f  A s  = 0 by ( 3 . 5 )  A s  and ( 3 . 6 ) )  market  and w i t h  u t i 1 i ty U (x)  =  A  where G to 0  A  A  i s g i v e n by ( 3 . 1 3 ) .  _  1  ( Q  Hence, i t  as a market r i s k t o l e r a n c e A t date 0 , a l l  Y  A  + X )  Y  i s a p p r o p r i a t e to  refer  parameter.  u n c e r t a i n t y about r e l a t i v e p r i c e s p  a r i s e s from e i t h e r u n c e r t a i n t y about the s t a t e s o r  g  i n date 1  uncertainty  about © , w h i c h , f o r computing p r i c e s , i s a " s u f f i c i e n t A  statistic"  f o r 0-j, . . . , 0 j . We can now compute the date 1 d e r i v e d u t i l i t y f o r w e a l t h  y^ , s  f o l d back to date 0 and s o l v e f o r date 0 demands (and hence p r i c e s ) i n the manner o f dynamic programming.  Substituting  (3.4)  and ( 3 . 1 1 )  i n t o the date 1 u t i l i t y and s i m p l i f y i n g y i e l d s the d e r i v e d  ^ •  E  c s c  0  A  + X  e  A  + p  cl  Y  n  s  (©•+y- )  As i n the d i s c u s s i o n o f Chapter 2 , suppose t h a t 0 are known to a l l  agents.  utility.  We w i l l  can be s u s t a i n e d i n e q u i l i b r i u m .  A  and hence p  check t o see t h a t i t s  g  revelation  Then the date 0 p o r t f o l i o  problem  becomes: Y'  max {f. subject  10  Lsn s  - 1  z  p  V  c sc  X  c f  + p  s  n  e  A  (e  i +  y )" i s  ,m. } 10  to: y. is  (3.1)  = f. + m. p io IO^S  7  f. + p m. io o io  (3.7)  y  = y. i J  T h i s i s an extended power u t i l i t y problem t h a t i s e x a c t l y analogous t o the date 1 problem when the p r o b a b i l i t y w e i g h t s are replaced";by n  n  s c  times the f a c t o r . i n l a r g e square b r a c k e t s above.  §  As i n the date 1 a n a l y s i s , e q u i l i b r i u m i s c h a r a c t e r i z e d b y : G^y. ( 3  '  1 4  >  m  i o  e  =  A  f l + pP  i = 1, . . . ,  o  [ A c] e  Zs ILs  E  0  I  c  sc  n  + X  Y  [ V s J ;  [ A c] e  s  s  E  c  sc  n  G  Cancelling factors in (e +p ) A  r e c a l l i n g that s n n O  (3.15)  p  O  V c  s  = n  X  c c f  This y i e l d s p i n v e r t i b l e then  a  Q  ?  + P  sj  substituting for p  g  from ( 3 . 1 2 ) and  W  H  c  r  s  A  yields:  Vrf  W W  + X  r  ( A 0  + X  c)  Y _ 1  as a f u n c t i o n o f 0^. fully  If  informing rational  (FRE) e x i s t s , as d i s c u s s e d i n Chapter 2 .  the f u n c t i o n  is  expections equilibrium  We have:  Z II C  c  (e  A +  X )  Y-1  dp  2  c  '0  " c,A«d £  ' fl 9  + X  c) " ( A Y  2  e  Wc (A c " 0  + X  d) "  0  Thus, p  Q  )Y  2E  d d ( A n  Q  + x  d)  Y _ 1  _  Y  C c«V d> - < A c » l X  2  = \ c , d ¥ d <V x c> Y " 2 ( A d ) " x  +X  + x  Y  2  X  e  + X  C(xc-xd)(xd-xc)]  < 0  i s an i n v e r t i b l e function of e and we have ft  Theorem 3.1 I f a l l agents have extended power u t i l i t y  .  . (3.9)  with decreasing absolute r i s k aversion and the market i s characterized by equations (3.1) to (3.7), then there e x i s t s a f u l l y informing r a t i o n a l expectations e q u i l i b r i u m (FRE) i n which agents can i n f e r aggregate r i s k preference  e ^ f r o m the  date,0_price  p and hence c o r r e c t l y i n f e r the p r i c e s p that would occur i f Q  g  state s occurred  (s = 1, ...., S).  This model i s somewhat akin to the Grossman [1976] model i n which i n d i v i d u a l s are endowed with exponential  u t i l i t y and have  independent i d e n t i c a l l y d i s t r i b u t e d observations about the next period mean value of a r i s k y asset's normally d i s t r i b u t e d return. Grossman showed that the market p r i c e i s an i n v e r t i b l e function of the sample mean observation and, hence, that a FRE e x i t s . One  feature o f h i s model that was  pointed out by Grossman-Stiglitz [1976]  and Kraus [1976] was t h a t , since prices convey a l l relevant information, i n d i v i d u a l s w i l l not c o l l e c t p r i v a t e information i f i t i s c o s t l y to obtain.  But i f they do not c o l l e c t p r i v a t e  information, i t w i l l never be impounded i n p r i c e s , so that prices w i l l not convey a l l a v a i l a b l e information a f t e r a l l . presented here does not have t h i s d i f f i c u l t y ,  The model  since agents must  assess t h e i r own u t i l i t y functions before coming to market, i n order to formulate t h e i r demands.  This e f f e c t i v e l y c o s t l e s s ,  heterogeneous information w i l l come to market.  '  C o r o l l a r y 3.2: Under the same hypotheses as f o r Theorem 3.1, but assuming a l l agents have  extended power u t i l i t y with i n c r e a s i n g  absolute r i s k aversion, v i z . (3.16)  u\(x) = - ( e x )  i = 1,  Y  r  I  where y > 1, there e x i s t s a f u l l y informing r a t i o n a l expectations from the date 0 p r i c e p .  e q u i l i b r i u m where agents can i n f e r Proof:  Q  By reasoning analogous to that o f Theorem 3.1, we have,  for example, 0. -y • m  - f j A  io  p  -= Lc ?c X, -Xj c (A0 n c'  o  Vc and  Y _ 1  v  ^o d0  > 0  5 )  ( e  A- c> X  Y _ 1  .  Q.E.O.  A  A natural question to ask i s how the r e s u l t s are a f f e c t e d by consumption a t dates 0 and 1, as well as date 2.  Suppose a l l  agents have i n t e r t e m p o r a l l y a d d i t i v e power u t i l i t y f u n c t i o n s , so that von Neumann-Morgenstern u t i l i t y becomes: Y"  1  (Vx  i 0  )  Y  + y"  1  1,^5115 ( e ^ ) *  where agent i consumes x ^ ,  +  x ^ and x^  Y'  $c  1  n ^c s sc ( i isc n  n  G  + x  2  r e s p e c t i v e l y a t date 0,  date 1 (state s ) , and date 2 (state ( s , c ) ) . This i s analogous to the previous problem with no intermediate consumption, where S+l  ) Y  s t a t e s , corresponding to date 0 and date 1 consumption have been added.  This has s i m i l a r aggregation and separation properties to the  market already studied, and i n appropriate s e t t i n g s , y i e l d s a FRE. At date 0 and date 1, markets separate into three assets:  date 0  or date 1 consumption, a r i s k l e s s asset and a r i s k y asset.  (In  f a c t , two assets w i l l do at each date, since the vector of agents' consumption good holdings i n &  1  w i l l be spanned by the vectors of I  6)  r i s k l e s s asset holdings and r i s k y asset holdings i n £ r e l a t i v e prices w i l l reveal 0^. w i l l have a special form: consumption at date 1.  .  Date 0  However, the date 0 r i s k l e s s asset  i t must provide one.unit of r i s k l e s s  Since the r e l a t i v e prices of these two types  of consumption ( i . e . , the rate of i n t e r e s t ) w i l l , i n general, be d i f f e r e n t in d i f f e r e n t states s at date 1, the date 0 r i s k l e s s asset cannot simply pay $1 at a l l date 2 states s: i t w i l l have a v a r i a b l e payoff i n d o l l a r terms.  I f markets are complete, then c l e a r l y such  an asset w i l l be provided ( i . e . , spanned), but i t s composition depends on the r e l a t i v e date 1 p r i c e s , which depend on 0^. not  Thus,;-  knowing 0^, i n complete markets an agent cannot compute which  weights to use on p r i c e s , i n order to compute the r e l a t i v e p r i c e of the r i s k y and r i s k l e s s assets, and hence i n v e r t to get 0^.  At  date 0, there would be S prices of the f i n a n c i a l s e c u r i t i e s r e l a t i v e to the date 0 consumption good.  This i s enough to y i e l d the s i n g l e  parameter Q ^ , under many c o n d i t i o n s , but a study of these does not appear to be very i n s t r u c t i v e . Intermediate labor income Another question i s how the r e s u l t s are a f f e c t e d i f date 1  (labor) income i s introduced f o r agent i .  S p e c i f i c a l l y , suppose  t h a t , at date 1, agent i receives income o f $L., independent of 1  s t a t e s, the value o f which i s revealed a t date 0 to agent i , but to nobody else.  Thus, date 1 aggregate wealth i s uncertain, a t  date 0, c r e a t i n g more uncertainty about date 1 p r i c e s .  We s h a l l  use the same notation as before with the f o l l o w i n g changes: The date 1-budget constraints become, ; f. + p is s  (3.17) ' v  M  .  m. = y. + L. is is i J  The date 1 market f o r the r i s k ! e s s asset c l e a r s when (3.18)  E . f. -|  = E.L. = L  is  11  The reason f o r having a nonzero aggregate supply of the r i s k less asset i s to ensure t h a t , by purchasing at date 0 the f r a c t i o n m  i>0  of the r i s k y asset, paying o f f  p  at date 1 i n s t a t e  s,  agent i i s only making a claim to the r i s k y market asset, rather than other agents' labor income. and using (3.18) y i e l d s P  s  which would be the case i f  = z.y  That i s , summing (3.17) over i is  rather than P  G  = ^y^ z  U  +  f.. = 0, which would be a e s t h e t i c a l l y ,  unappealing.  In t h i s s e t t i n g , the f o l l o w i n g holds:  Theorem 3.3:  Under the same assumptions as f o r Theorem 3.1 (ex-  tended power u t i l i t y , e t c . ) , but s u b s t i t u t i n g (3.17) and (3.18) f o r (3.4) and (3.5), r e s p e c t i v e l y , where L. i s agent i ' s date 1 labor income, known to only him a t date 0, there e x i s t s a f u l l y informing r a t i o n a l expectations e q u i l i b r i u m in which the date 0 p r i c e p reveals 0^ + L, which allows computation o f date 1 prices p Proof:  $  Q  .  Using the techniques of the proof of Theorem (3.1), one  can show that:  _ Vsc c  < A  + L  +  '  (G L  +  X  P s  and  E n c  6  s  A +  X  c) " Y  ]  X r  ]  c  0- + L. + y. ; = -J ! Li ns 0 + L + p :  m  A  s  A t date 1 i n state s , the u t i l i t y f o r f i n a n c i a l wealth y..  is,  f o r agent i , Y  (3.19)  z n G  e s c  A  •A e  If  agents know 0  +  L + X  +  L  + p  ( i 0  +  L  i  ^is)  is  (e. + L + x J Y _ 1 A ' ' c (e + L X )Y-1  'c"c c o " ^ n c  +  s  + L , the date 0 p r i c e  A  A  p  Y  c  V U  c  u  A  A  +  C  and  io Thus, a l l  +  0. L. + y . 0 i. + L I + p I A  Q  p r i c e s are a f u n c t i o n of 0  A  + L i n the same way t h a t  p r i c e s i n Theorem 3.1 were a f u n c t i o n o f Q^. 0  A  i n Theorem 3 . 1 , p  reveals ©  Q  + A  J u s t as p  i  reveals  L h e r e , and a FRE e x i s t s .  An i n t e r e s t i n g v a r i a t i o n on t h i s market s t r u c t u r e that L  Q  Q.E.D  i s to suppose  i s r e v e a l e d to agent i a t date 1 o n l y , and i s random a t  date 0 , denoted by  .  Suppose, f o r s i m p l i c i t y ,  t h a t L = E^L^ i s  not random, b u t known to a l l  agents a t date 0 .  c o u l d a r i s e , f o r example, i f  agents a r e s t e v e d o r e s who  to a h i r i n g h a l l e v e r y p e r i o d .  Such a s i t u a t i o n  Workers a r e a s s i g n e d to  randomly, s i n c e t h e r e i s not enough work f o r a l l .  report jobs  However, a l l  workers know beforehand the t o t a l amount o f l a b o r to be s u p p l i e d c o n t r a c t u a l l y by the u n i o n .  T h i s minor change p r e v e n t s the e x i s t e n c e o f a FRE, .although the economy may admit a r a t i o n a l not f u l l y r e v e a l i n g . the  expectations equilibrium that  A FRE would have agents h o l d i n g a t date 0  fraction:  9-+L.+y. m. 'io  e  A +  L  + P o  o f the r i s k y a s s e t (from the p r o o f o f Theorem 3 . 3 ) . a l l o w them to compute random demand m. although t h i s  Q  ,  since  information  i s not . a v a i l a b l e to any agent a t date 0 . In t h i s market s t r u c t u r e ,  date 1, s t a t e  i s known t h e n .  t i o n s over s t a t e  the  to g e n e r a t e the whole i n f o r m a t i o n a - a l g e b r a L...,  r e p r e s e n t s agent i ' s i s  I t would a l s o  + L from p , so i t would r e q u i r e  Hence a FRE cannot e x i s t .  y  s  and  s  still  derived u t i l i t y for  wealth  A t date 0 , agent i must t a k e e x p e c t a s  s  =  fe +L+xJ A  s c s sc  (e.+L.+y. ) i i •'is'  Y  v  L + p  s  e.-, w i t h r e s p e c t to t h e random v a r i a b l e s e ,  a t - d a t e . 0;'.  Tf a r a t i o n a l  may be assumed t h a t the 1 .  0  s  H  Q  and  u n c o n d i t i o n a l l y and c o n d i t i o n a l on  expectations  equilibrium  are independent o f e , y .  and  f l  A  1  $  f-j P  p r o v i d e s no i n f o r m a t i o n about L . , s i n c e no agents have  i n f o r m a t i o n about  p ,  + S  p_, y . _ a n d , e . .  f l  exists, it  -j  e.i ',po  The e x p e c t a t i o n o p e r a t o r E i s the e x p e c t a t i o n c o n d i t i o n a l on p  Q  m  The e x p e c t e d u t i l i t y becomes:  JL£„iL I L „ E  V  (3.19)  , as w e l l as o v e r e ^ , p , and y ^  s i n c e a FRE d o e s n ' t e x i s t .  Note t h a t p  is  P q  .  The randomness o f  I S  serves  to make markets i n c o m p l e t e , t h e r e b y d e s t r o y i n g the s e p a r a t i o n and  43. aggregation p r o p e r t i e s .  A n o t h e r s i m p l i f y i n g assumption may be  c a l l e d an " i n f o r m a t i o n - t a k i n g " a s s u m p t i o n about a g e n t s , analogous t o the p r i c e - t a k i n g assumption used i n the t h e o r y o f markets.  competitive  That i s , one can assume, as an a p p r o x i m a t i o n from some  )/(e +L+p )  law o f l a r g e numbers, t h a t the d i s t r i b u t i o n o f (e.+L+x is  independent o f t h a t o f 6 ^ , even though 0  if  t h e r e a r e enough a g e n t s , agent i , w i l l  about p „ from h i s own e . so t h a t p  n  will  =  A  fl  1  C  M  S  6^.  That i s ,  n o t make any i n f e r e n c e s depend on e . o n l y through 6„  Computing the f i r s t o r d e r c o n d i t i o n s and s u b s t i t u t i n g  for  from the budget c o n s t r a i n t y i e l d s ( w i t h endowment y^ = m^P )> i n 0  the g e n e r a l  case, rt..  „  A  6  Ei  z z n n  s c s sc  + L + X  c  \y  P (6 +L. m.p m (p -p )) s  l A 9  e  s c s sc  >  A  6  + L + p  + L + X  A +  +  0 +  i 0  s  Y-1  0  c  fe . + L . + m . p „ + n i . „ ( p ^ - p „ ) l  n  1 1  = 1.  T h u s , t h e r e a r e I+l  = 1,...,I). conditions  1 1 0  on the endogenous v a r i a b l e s m^  Q  and p  Q  t h a t d e f i n e them, s u b j e c t  a c o n s t r a i n t on the exogenous v a r i a b l e s , for  i  S  = i . e . and i , . m .  M '•  e. , m  Y-1  (i We a l s o have e  P , 0  si  L P +  i  and  .  to  By s u b s t i t u t i n g  from the budget c o n s t r a i n t , we deal w i t h one market o n l y so  W a l r a s ' Law does not reduce the i n f o r m a t i o n c a r r y i n g c a p a c i t y o f p r i c e s . The e x i s t e n c e o f a r a t i o n a l  e x p e c t a t i o n s e q u i l i b r i u m i n t h i s s e t t i n g has  not been e s t a b l i s h e d , b u t a s s i g n i n g p r o b a b i l i t y d i s t r i b u t i o n s  to  and L. would r e s u l t i n a w e l l - d e f i n e d Banach space f i x e d p o i n t p r o b l e m , which may be s t u d i e d n u m e r i c a l l y by computer.  >  FRE's w i t h o t h e r l i n e a r r i s k t o l e r a n c e u t i l i t y A natural  classes  q u e s t i o n i s whether a FRE e x i s t s when agents have  u t i l i t y functions  i n the o t h e r c l a s s e s e x h i b i t i n g  linear  risk  t o l e r a n c e , namely the extended l o g and e x p o n e n t i a l c l a s s e s . answer i s a f f i r m a t i v e , Theorem 3.4  o f the  as i n the n e x t two theorems:  Suppose the market s t r u c t u r e  e x c e p t t h a t agents a l l  The  o f Theorem 3 . 3 h o l d s ,  have extended l o g u t i l i t y f o r date 2 w e a l t h  form U .(x) = li) (e.j+x) n  Then a FRE e x i s t s i n which agents can i n f e r e +L from p , and > A  hence can a l s o i n f e r the p r i c e s t h a t w i l l Proof:  o b t a i n a t date 1.  The f i r s t o r d e r c o n d i t i o n s a r e the same as f o r the power  u t i l i t y c l a s s o f Theorem 3 . 1 , where y = 0, a n parameter v a l u e f o r power u t i l i t y  inadmissible  :  (igiving constant u t i l i t y ) .  p r o o f o f Theorem 3.1 a p p l i e s w i t h y = 0 , needing o n l y  elementary  modifications. Theorem 3.5  The  Q.E.© Suppose the market s t r u c t u r e  3.4 h o l d s , e x c e p t t h a t a l l  o f Theorems 3 . 3 and  agents have e x p o n e n t i a l  u t i l i t y of  the  form U .(x) = - exp(-G n  Then a FRE e x i s t s  X). .  . i n which agents can i n f e r  and hence can a l s o i n f e r Remark:  i  o E ( Z ^ .  the p r i c e s t h a t w i l l  -, -1 ') from p  Q  o b t a i n a t date 1.  A t date 0 , agents cannot and need not make any i n f e r e n c e s  about aggregate date 1 l a b o r income L .  Essentially, this  because the aggregate i n v e s t o r has e x p o n e n t i a l  arises  u t i l i t y and hence  constant absolute r i s k aversion.  Wealth does not a f f e c t choices  among gambles, and hence does not a f f e c t the marginal rate of s u b s t i t u t i o n between the r i s k y and safe assets. Proof:  At date 1 i n s t a t e s, the f i r s t order conditions f o r  agent i y i e l d : exp v  -0.  exp  -0.  E„IL„ .X.  c sc  p  s  c  =  c sc  £ =  ( f . +m. "I  c SC C n  ( f . +m. X J i s i s c'  1  X  e X  Vsc  P  IS  (- i i S  \\  i  V  0  e x p  (  L  0  m  m  is  X ) c  IS  Note that the f a c t o r i n f. i s j u s t a wealth e f f e c t and drops out is The f i r s t order conditions are the same f o r a l l agents when i is the sharing r u l e :  0  m  =  c o n s t a n t  =  Since z. m  1  = 1, we have  is  -= 9 —  m.  is  0.  and the p r i c e becomes £  c sc c  z  c sc  n  e P>  X  n  x  e x  P  0 X  c>  (- c^ 0X  A f t e r computing the derived u t i l i t y f o r date 1 wealth  +  y , i $  the date 0 f i r s t order c o n d i t i o n becomes, f o r agent i , >  . s c s sc Z  E  ° ~Vc  n  n  n  s sc n  ^ ^ s ^ c ^  e x p  e x  s, P Kj  l <VVJ  P  c l e a r s with the sharing r u l e :  ' i  Q  and L  ^io^i^ioPs^  e x  e  Once again, the f a c t o r s i n f .  io  p  e x p  i  i-¥ io f  + L  i  + m  .io s)) p  cancel and the market  .  and p r i c e : V c V s c P  °  n  =  n  c  w.r.t  Q  e  e x p  e x  (  -  6  X  c  }  P(-ex ) c  X_ exp  c c E n  p  s  Vc s sc  EH  Differentiating  p  c  (r&  Xj  c ,  r  -.  exp(-ex )  c  G  and r e - a r r a n g i n g terms i n t o a sum  o f squares as i n Theorem 3.1 y i e l d s : dP  0  de  Hence  a FRE can be s u s t a i n e d , s i n c e o n l y 0 and not L i s needed  to compute p .  Q.E.D  $  Rollover of portfolios  a t date 1  A r e - e x a m i n a t i o n o f the p r o o f o f Theorem 3.5 y i e l d s a d i s t u r b i n g o b s e r v a t i o n , namely, t h a t : m.  = 0/0.  10  Thus agent i w i l l  (i = l ,  ...,  I;  s = 1  S).  o f the r i s k y a s s e t and the remainder o f h i s  w e a l t h i n the s a f e a s s e t . S  TS  t r a d e a t date 0 t o a p o r t f o l i o c o n s i s t i n g o f  the p r o p o r t i o n 0/0^  P  = m. 1  A t date 1 i n s t a t e s ,  t h i s w i l l be worth  (0/0.j) which i s j u s t enough to purchase the same f r a c t i o n  of  the aggregate date 1 r i s k y a s s e t , which a l s o happens to be h i s optimal h o l d i n g .  The date 0 r i s k ! e s s a s s e t h o l d i n g s f .  to l a b o r income L. to g i v e the o p t i m a l f o r d a t e 1.  Q  are added  r i s k l e s s asset holdings  f..  T h u s , a t date 0 , agent i may view the r i s k y a s s e t  not as an a s s e t paying p a t date 1 i n s t a t e s , but as an a s s e t g  paying X  c  a t date 2 i n s t a t e ( s , c ) . . .  This w i l l  a l l o w him to  to an o p t i m a l demand h o l d i n g m ^ , which he merely needs to  trade  re-invest  mechanically ("rollover") a t date 1 i n t o the same amount of the r i s k y asset.  I f a l l agents do t h i s , date 0 prices w i l l be the  same as i f the date 0 problem i s viewed as a conventional stage problem.  two  Using t h i s p o r t f o l i o r o l l o v e r technique, agents  need not p r e d i c t date 1 prices p , and hence have no need to know e.  The machinery used to e s t a b l i s h the existence of a FRE i s  not r e a l l y needed because agents can achieve optimal using the r o l l o v e r technique.  holdings  Interpreted broadly, t h i s r o l l o v e r  economy o f f e r s a FRE i n the sense that agents can behave as i f they were f u l l y informed.  A l t e r n a t i v e l y , the market may be domina-  ted by .investors^with-.exponent,ial:-utiTity', while an' i n f i n i t e s i m a l investor with a d i f f e r e n t u t i l i t y function w i l l not choose to rollover  holdings, but w i l l desire to trade a t date 1.  an i n v e s t o r , i t i s important to i n f e r the aggregate r i s k  For such aversion  that sets p r i c e s . The question a r i s e s as to whether or not the same r o l l o v e r technique works f o r extended power and l o g u t i l i t y economies. For these economies, the optimal  r i s k y asset holdings a t date 1  (state s) and date 0 are, r e s p e c t i v e l y :  m  V ^ i s  is  e L + p A+  and,  9  i  Q  A  m.  + L  s  i yj +  o  Expanding m^ » we f i n d that: s  + L  +  p  o  —— -—!—Ly_£_^  by  l  e  A  +  L +  p  (3.7)  s  m. (e.+L+p + - ) 1 0  "  H  e +L+ A  M  P  P  b  u  s u b s t i t u t i n g m.  1 0  from  P  above.  s  m.  10  Thus, the r o l l o v e r s i m p l i f i c a t i o n i s p r e s e n t i n the power and l o g u t i l i t y e c o n o m i e s , as w e l l .  The r o l l o v e r a l g o r i t h m a r i s e s  because h o l d i n g s i n the market p o r t f o l i o over i n t o i d e n t i c a l  a t date 0 can be r o l l e d  h o l d i n g s i n any s t a t e a t date 1.  holds f o r the r i s k l e s s a s s e t :  The same  $1 a t date 0 b r i n g s $1 i n whatever  s t a t e s o c c u r s a t date 1, which b r i n g s $1 i n whatever s t a t e c o c c u r s a t date 2 . the date 1  (With l a b o r income, the agent merely counts  as p a r t o f "human w e a l t h " a t date 0 . )  Hence, at  date 0 , agents can e f f e c t i v e l y buy c l a i m s to the date 2 r i s k l e s s a s s e t and market p o r t f o l i o . they would purchase i f  S i n c e these a r e the o n l y s e c u r i t i e s  p r e s e n t e d w i t h a complete s e t o f date 2  c o n t i n g e n t c l a i m s , they e f f e c t i v e l y  f a c e a complete market  in  date 2 c o n t i n g e n t c l a i m s , a l t h o u g h a t date 1 they r e v i s e  their  estimates  final  states.  of  the p r o b a b i l i t i e s o f the o c c u r r e n c e o f the  H i r s h l e i f e r [1971] and M a r s h a l l [1974] have shown t h a t  the c o n t r a c t c u r v e o f an exchange economy w i t h complete markets does not depend on the s t a t e p r o b a b i l i t i e s , so t h a t a t date 0 , agents t r a d e to the c o n t r a c t c u r v e , and do not r e - t r a d e a t date 1. even though they r e v i s e t h e i r p r o b a b i l i t y  i n f o r m a t i o n a t date 1.  Note t h a t t h i s r o l l o v e r f e a t u r e i s not r e l a t e d to the myopia o f Mossin [1968] f o r l i n e a r r i s k t o l e r a n c e u t i l i t y f u n c t i o n s . H i r s h l e i f e r - M a r s h a l l r e s u l t i s i n a general e q u i l i b r i u m  The  setting,  w h i l e myopia i s r e l a t e d to a s i n g l e i n d i v i d u a l ' s p o r t f o l i o demand. By s t a y i n g on the c o n t r a c t c u r v e and merely r o l l i n g o v e r portfolios  their  o f r i s k y and r i s k i ess a s s e t s , agents are f o l l o w i n g a  s t a t i o n a r y i n v e s t m e n t p o l i c y even though y i e l d d i s t r i b u t i o n s may not be s t a t i o n a r y  (e.g., if  the p r o b a b i l i t i e s o f some s t a t e s become  z e r o , the number o f e f f e c t i v e s t a t e s changes and y i e l d must change, no m a t t e r how p r i c e s move).  distributions  This provides a counter-  example to M o s s i n ' s [ 1 9 6 8 , p.122] c o n t e n t i o n t h a t agents w i l l s t a t i o n a r y investment p o l i c i e s only i f y i e l d d i s t r i b u t i o n s  have  are  stationary. t h i s r o l l o v e r f e a t u r e cannot be a v o i d e d i n a s t a t e p r e f e r e n c e s e t t i n g w i t h o u t l o s i n g the c l o s e d form s o l u t i o n s f o r p r i c e s .  The  c l o s e d form p r i c e s a r i s e from the a g g r e g a t i o n and s e p a r a t i o n i n l i n e a r r i s k t o l e r a n c e u t i l i t y c l a s s and t h e e f f e c t i v e l y markets a t dates 0 and 1.  complete  Without complete m a r k e t s , a g g r e g a t i o n  f a i l s , but w i t h complete m a r k e t s , the H i r s h l e i f e r - M a r s h a l l obtains.  the  result  T h i s m o t i v a t e s the model o f the n e x t s e c t i o n i n which  date 0 markets a r e n o t c o m p l e t e , but s e p a r a t i o n o b t a i n s because r e t u r n s a r e n o r m a l l y d i s t r i b u t e d and p r i c e s are r e a d i l y computed by u s i n g c o n s t a n t a b s o l u t e r i s k a v e r s i o n ( e x p o n e n t i a l )  utility.  50.  A model i n which date 0 p r i c e s r e v e a l aggregate p r e f e r e n c e s and agents re-balance portfolios  i n the i n t e r m e d i a t e  period  As i n t h e p r e v i o u s s e c t i o n s , c o n s i d e r a pure exchange economy w i t h I agents and t h r e e dates ( 0 , 1, 2 ) .  A t date 0 t h e r e a r e two s e c u r i t i e s  which have j o i n t l y n o r m a l l y d i s t r i b u t e d date 1 p a y o f f s r r  w1  w1  ~ N  w.  a\  0  0  al  A t date 0 agent i s e l e c t s a p o r t f o l i o v e c t o r ( c t ^ - j , r e t u r n s a t date 1 o f  = a.^W-j + a W .  (3.20)  y  where y  i 2  = a„  i Q  0  i s h i s i n i t i a l endowment and p  i Q  a s s e t ( w i t h the f i r s t as n u m e r a i r e ) .  (3.21)  - l  2  a n c  * realizes constraint  i 2  i s the p r i c e o f the second  Q  Markets a t date 0 c l e a r when  •  f  il  a  -j )'  He f a c e s the budget  2  + P a  a  1  \  1  v.• i 2 J a  A t date 1 , t h e r e i s a r i s k l e s s a s s e t w i t h r e t u r n R and a r i s k y a s s e t w i t h a date 2 p a y o f f  of V ~ N(V,  Agent i r e a l i z e s date 2 w e a l t h y where e asset. (3.22)  i R  i 2  S  Oy)  of  3  i R  R + 3  i s h i s l e n d i n g / b o r r o w i n g and e  He f a c e s the budget y  i  l  i v  i v  constraint = 6  i R  +  P^iv  V i s h i s h o l d i n g o f the  risky  where p-j i s the p r i c e o f the (second p e r i o d ) r i s k y a s s e t ( w i t h t h e r i s k l e s s a s s e t as n u m e r a i r e ) .  The market c l e a r s when  •  0  I  (3.23)  \  Individual  1  "1V  »  J  4  i has u t i l i t y f o r f i n a l w e a l t h o f • - e x p ( - e . y ) •  A  t  d  a  t  e  i 2  he maximizes e  i  - E e x p ( - e ( / 3 . R ^ . V ) : . ) =• -exp(-0 (/3 R+/3. V)+ — i  R  s u b j e c t to ( 3 . 2 2 ) .  v  i  S o l v i n g (3.22)  j e c t i v e y i e l d s the f i r s t o r d e r  iR  v  for , 0 .  2 2 ^. a ) v  v  and s i m p l i f y i n g the o b -  R  condition  e e ^ V - p ^ - e^a ).exp(-"e.(0 &tf 2  i R  1 v  2  V ) + ~$\^\)  = 0  .  Hence V-P{R  (3.24)  ^iv  - <  Summing o v e r i and n o t i n g (3.23)  ( 3  -  p, -  2 5 )  where 0 = (£ 0 ~^)~^ i i  .  yields  •*>  T h i s g i v e s the c a p i t a l a s s e t p r i c i n g model  (CAPM) p r i c e o f the r i s k y a s s e t . Using (3.24) and ( 3 . 2 5 ) , t h e d e r i v e d u t i l i t y f o r date 1 w e a l t h y ^ l becomes -expt-G^y^)  exp(-ia9 0 ) . 2  2  1  2 Assume t h a t R i s known and f i x e d b e f o r e h a n d , and t h a t o-y  is  n o n s t o c h a s t i c so t h a t the second f a c t o r i n the above u t i l i t y f u n c t i o n i s a s c a l i n g f a c t o r t h a t may be i g n o r e d . date 0 , i n d i v i d u a l  Thus, at  i maximizes  0 R 2  e x p f - e ^ R y ^ ) = -exp(-'e.R(a W.| +  -E  i1i  s u b j e c t to the budget ( 3 . 2 0 ) . the f i r s t  D i v i d e by  =  1  " h  +  0  i  R (  - o p  and sum over i ,  +  p  o  W  t  0  g  1  W )  +  2  2  Substituting for  from the b u d g e t ,  2 2  2  io '  P  o i2 a  }  a  l  2  +  a  i2 2 ) a  2  =  0  n o t i n g from ( 3 . 2 0 ) and (3.21)  + 9 R (-P (l P )o-, +  2  0  -  W  that  0  2  + PQO-,  2  + a ) 2  2  = 0  2  L  P C ) 2  0  7 , W , a-j  ( y  t  "  0  R(o  e  - W  IPQ  V o  so t h a t e = -  Since  I 2  2  i 2 2 2 2 — — ( a ^ ^ + a ^ a ))  o r d e r c o n d i t i o n s become, a f t e r s i m p l i c a t i o n :  Vo  Vio  a  -  7 , a  2  and R a r e known, one can compute e knowing p , Q  so t h a t a F R E e x i s t s and s i n c e date 0 markets a r e i n c o m p l e t e , agents must r e - b a l a n c e t h e i r p o r t f o l i o s a t date 1. Theorem 3.6  This y i e l d s the  following  In a two p e r i o d ( t h r e e d a t e ) economy, where the second  p e r i o d market c o n s i s t s o f a n o r m a l l y d i s t r i b u t e d r i s k y a s s e t and a r i s k l e s s a s s e t ( w i t h known i n t e r e s t r a t e ) , t h e f i r s t  p e r i o d market  c o n s i s t s o f two n o r m a l l y d i s t r i b u t e d r i s k y a s s e t s , and a l l agents have e x p o n e n t i a l u t i l i t y , as d e s c r i b e d i n t h i s s e c t i o n , market p r i c e s  depend on the aggregate r i s k a v e r s i o n parameter 0 which i s r e v e a l e d by d a t e 0 p r i c e s .  Hence a f u l l y  informing r a t i o n a l  expectations  (FRE) e x i s t s , even though date 0 markets are i n c o m p l e t e and agents must r e - b a l a n c e t h e i r p o r t f o l i o s  a t d a t e 1.  The t r a c t a b l e computations i n Theorem 3.6 r e s u l t e d from the p o r t f o l i o  partly  s e p a r a t i o n t h a t was induced by the n o r m a l l y  distributed portfolio  returns.  Date 0 markets do not have a r i s k -  l e s s a s s e t and hence are i n c o m p l e t e so t h a t the e x p o n e n t i a l function  itself  i s not enough to ensure p o r t f o l i o  separation.  suggests t h a t the theorem may a l s o extend to o t h e r u t i l i t y f o r which the  has s u p p o r t on the whole r e a l l i n e . )  This  functions  u t i l i t y of negative wealth i s w e l l - d e f i n e d .  normal d i s t r i b u t i o n  utility  (The This  e l i m i n a t e s u t i l i t y f u n c t i o n s such as extended l o g and power w i t h tional  o r the ' i n v e r s e o f even e x p o n e n t s .  U!j(x) = Y ^ ( Q i  +  X  )  Y  W  I  T  N  Y  =  m  For-power u t i l i t y o f the  T h i s s e p a r a t e s but does not aggregate i n  date 0 m a r k e t , and does aggregate i n the date 1 market date 1 p r i c e s depend on a different  two  the  ^ H e n c e ,  = -j -j> but the date 0 p r i c e depends on z  f u n c t i o n o f (e-j,  0  0j).  In t h i s c a s e , i t  is  doubtful  whether a FRE e x i s t s , so t h a t Theorem 3.6 may not be r o b u s t w i t h r e s p e c t to r e l a x a t i o n o f the e x p o n e n t i a l u t i l i t y c l a s s a s s u m p t i o n . This l i n e of reasoning a l s o suggests t h a t i t to make s t r o n g ! - d i s t r i b u t i o n a l ' a s s u m p t i o n s s e p a r a t i o n , as i n Ross [ 1 9 7 8 b ] ,  i s not adequate  t h a t ensure  portfolio  to ensure t r a c . t a b i l i t y , . s i n c e the  a g g r e g a t i o n r e s u l t s must a l s o o b t a i n to a c h i e v e the parsimony o f parameters.  form  / ( 2 n + 1 ) ; m,n i n t e g e r s , the e x -  p e c t e d u t i l i t y i s w e l l - d e f i n e d and can be computed as a sum o f gamma f u n c t i o n s .  irra-  The m o t i v a t i o n f o r the assumptions o f Theorem 3.6 came from a d e s i r e to use c a p i t a l a s s e t p r i c i n g model (CAPM) r e s u l t s w i t h and w i t h o u t a r i s k i ess a s s e t (and h e n c e , w i t h and w i t h o u t markets) to get two fund s e p a r a t i o n .  complete  The two fund s e p a r a t i o n r e -  s u l t s w i t h o u t a r i s k l e s s a s s e t f o l l o w from B l a c k [ 1 9 7 2 ] .  For the  CAPM, t h e market p r i c e o f r i s k can be e x p r e s s e d i n terms o f each agent's "global r i s k aversion" E UV'tx) E U.'(x) (cf.  Rubinstein [1973]).  u t i l i t y , exponential  Since t h i s i s constant only f o r  exponential  u t i l i t y was chosen f o r the m o d e l .  I t would a p p e a r , t h e n , t h a t the o n l y r e a l g e n e r a l i z a t i o n t h a t t h i s model admits ( t h a t r e t a i n s the p a r s i m o n i o u s a g g r e g a t i o n and separation properties) normal a s s e t s .  The r e s u l t s a r e s t r a i g h t f o r w a r d ,  o b t a i n s even w i t h o u t It  i s the g e n e r a l i z a t i o n to s e v e r a l  i s even hard to  multivariate  since separation  the r i s k l e s s a s s e t (as i n the B l a c k [1972] CAPM). i n c o r p o r a t e i n c o m p l e t e markets i n the second  p e r i o d as w e l l as the f i r s t , s i n c e then the d e r i v e d u t i l i t y f o r date 1 w e a l t h has a q u a d r a t i c f u n c t i o n i n the e x p o n e n t , y i e l d i n g a b l i s s p o i n t o f maximum g l o b a l u t i l i t y . ingful  without  Such an economy would not be mean-  f r e e d i s p o s a l beyond the b l i s s p o i n t , and the  r e s u l t i n g d e r i v e d u t i l i t y becomes i n t r a c t a b l e .  Conclusion The models o f t h i s c h a p t e r were f o r m u l a t e d to study whether c u r r e n t p r i c e s can r e v e a l enough i n f o r m a t i o n about p r e f e r e n c e parameters to r e s o l v e some o f the u n c e r t a i n t y about p r i c e s the n e x t p e r i o d .  For s i m p l i c i t y , a l l  in  the models were c o n s t r u c t e d  so t h a t o n l y an a g g r e g a t e p r e f e r e n c e parameter had t o be c o n v e y e d . Chapter 4 d e a l s w i t h the o t h e r major exogenous f a c t o r  (besides  p r e f e r e n c e s ) t h a t a f f e c t s demands and p r i c e s : the a l l o c a t i o n o f endowments.  S i n c e , by d e f i n i t i o n , a g g r e g a t i o n o b t a i n s when the  a l l o c a t i o n o f endowments does not a f f e c t p r i c e s , t h i s f a c t o r must be s t u d i e d i n a n o n - a g g r e g a t i o n s e t t i n g .  F o o t n o t e s t o Chapter 3 .  1.  One may r e g a r d  as a r i s k t o l e r a n c e p a r a m e t e r , s i n c e  (3.9)  has d e c r e a s i n g a b s o l u t e r i s k a v e r s i o n , so t h a t i n c r e a s i n g 8. i s e q u i v a l e n t t o i n c r e a s i n g w e a l t h and hence d e c r e a s i n g absolute r i s k a v e r s i o n , or increasing r i s k 2.  It  tolerance.  i s v a l u a b l e t o e s t a b l i s h t h a t t h e r e i s a unique  system t h a t e q u i l i b r a t e s  the economy, so- t h a t ( 3 . 1 2 )  r e p r e s e n t the e q u i l i b r i u m p r i c e s y s t e m .  If  p r i c e s are r e p r e s e n t e d by ( 3 . 1 2 )  does  p r i c e s are  u n i q u e , agents must have some way o f knowing t h a t  p r i c e system.  price  not  equilibrium  r a t h e r than some o t h e r  In economies where a l l  agents have u t i l i t i e s  from one c l a s s t h a t a g g r e g a t e s ( t h a t i s , f o r which aggregate excess demand f u n c t i o n s are u n a f f e c t e d by  redistributions  o f endowments amongst i n d i v i d u a l s ) , aggregate excess demands and p r i c e s a r e formed as though a l l w e a l t h were endowed upon a s i n g l e agent ( w i t h s u i t a b l e aggregate p r e f e r e n c e s ) , a n d , in effect  become "one household e c o n o m i e s . "  If  in  addition,  demands a r e s i n g l e - v a l u e d , s u i t a b l y c o n t i n u o u s and bounded from b e l o w , a o n e - h o u s e h o l d economy e q u i l i b r a t e s w i t h a unique p r i c e v e c t o r (cf_. Arrow-Hahn [ 1 9 7 1 , pp. In f i n a n c e , p r e f e r e n c e s aggregate i f f exponential, or a l l or a l l  217-220]).  u t i l i t i e s are  all  extended power w i t h the same exponent  extended l o g , as used i n C h a p t e r s 3 and 5 ( cf_.  R u b i n s t e i n [1974] and Brennan-Kraus [ 1 9 7 8 ] ) . yield sufficiently  These u t i l i t i e s  w e l l - b e h a v e d excess demands so t h a t p r i c e s  are unique i n complete markets o r markets where t h e r e  is  p o r t f o l i o s e p a r a t i o n y i e l d i n g e f f e c t i v e l y complete markets (and hence a d m i t t i n g  aggregation).  A n o t h e r way t o c o n s i d e r t h i s , suggested by Stephen R o s s , i s to s p l i t a l l w e a l t h between two  identical  aggregate  i n v e s t o r s and note t h a t , i n the Edgeworth box f o r a complete s e c u r i t i e s m a r k e t , the c o n t r a c t c u r v e i s , by symmetry, a l i n e .  T h i s l i n e i s a l s o an Engel c u r v e , and  s i n c e the p r e f e r e n c e s t h a t l e a d to a g g r e g a t i o n i n  finance  are a l l  » ),  homothetic  (through some p o i n t , perhaps -  the i n d i f f e r e n c e c u r v e s a l l  c u t the E n g l e - l i n e a t the same  a n g l e so t h a t o n l y one p r i c e  h y p e r p l a n e can e q u i l i b r a t e  the economy f o r any one s e t of endowments and p r e f e r e n c e s . :  Agent 2  Agent 1  Figure 3 . 1 .  Edgeworth box w i t h l i n e a r c o n t r a c t c u r v e , homothetic i n d i f f e r e n c e c u r v e s and endowment e .  C h e c k i n g t o - s e e whether the bases o f the exponents used throughout  t h i s a n a l y s i s a r e p o s i t i v e , note t h a t we must  assume 8 . + X _ £ 0 , c = 1 , . . . , C . (s = 1 , . . . , S )  and p  > 0.  By ( 3 . 1 2 )  A l s o , from ( 3 . 1 2 )  (e.+x )  z n e.+  and ( 3 . 1 5 ) , p >0  = ,  P c  A  c s c  and  (3.15),  Y  -j  c  En (e +X ) " c sc A c Y  ]  >  0  fl  (0 +X )  zji and  e + fl  P  =  o  c  f l  c  H  Y  r  >  c  ^ ( e . + X J ^ c c A c Also, in  0  1  equilibrium, i.+ x . = e . + f . ' „ + m. X. i isc i is i s c (0 +X  )  A  7  = (o.+ry. ) ,/X rs'  •i  This i s p o s i t i v e i f f  0.+ 1  y. J  0.+ y  is  which i s p o s i t i v e i f f K  this e  i  (e +p ) A  > 0,  T.  " V  • •  s  By s i m i l a r r e a s o n i n g ,  = (6.+ y.) v  ....  *  w  (  E  + A  P  Q  )  e . + y . = 0 . + m.p i -\ i i o J  > 0.  Since p  0  > 0,  i s p o s i t i v e under a wide range o f c i r c u m s t a n c e s — e . g .  1 0>  171  -j  >  0-  Note t h a t one r a t i o n a l e f o r extended power  and l o g u t i l i t y i s t h a t - 0 . > 0 i s a minimum s u b s i s t e n c e l e v e l o f c o n s u m p t i o n , so "that x ^ the s u b s i s t e n c e l e v e l .  s c  i s consumption beyond  This r a t i o n a l e f a i l s  if  e.. > 0 .  F i n a l l y , we note that f o r c e r t a i n parameter values, some agents may go bankrupt i n some states ~ x.  s 0 or y.  f S0  i . e . , have  < 0 f o r some i , s, c. As long as agents are  IS  not allowed to d e f a u l t a t date 2 i f x  < 0  i s c  (e.g., negative  consumption of f i n a n c i a l wealth i s f e a s i b l e i f agents have human wealth that they can use to pay o f f debts a t date 2 ) , then, even i f y  i s  < 0 a t date 1, they w i l l choose to hold a  p o r t f o l i o as indicated by the sharing rules (and an " i n t e r i o r " optimum f o r t h e i r f i r s t order conditions) rather than d e f a u l t at date 1, so long as e^y... > 0, as indicated i n Figure.-3.2 below.  (-e.,0) Figure 3.2  y <0 is  (0,0)  f  i  s  I n t e r i o r optimum  The i n d i f f e r e n c e curves f o r f . and m. IJ  induced by the s t r i c t l y I •)  concave von Neumann-Morgenstern u t i l i t y cross the m  is  a x i s , but  do y i e l d i n t e r i o r optima s a t i s f y i n g the standard Lagrange equations. To see t h i s , imagine that the 0.. i s human wealth to be received -1 Y at date 2 i n the proof of Theorem (3.1), and u t i l i t y i s -y ( x ) '  60.  where  x  i s t o t a l f i n a l wealth. See also Theorem 3.3.  I am endebted t o S. Grossman f o r reminding me of t h i s nice feature of the model. Now, we require e >X ,c = 1,...,C, f o r p and p to be A c ; o s n  n  real.  As i n footnote 3, t h i s ensures that the bases o f  exponents w i l l be p o s i t i v e , as long as e.>y.,i = 1,...,I. For example, i f asset markets are complete at date 1 i n state s, agent i can be viewed as facing r i s k y returns with S+l; states (consumption and the s states).  previous r i s k y  There i s no problem r e s u l t i n g from the f a i l u r e  of the p r o b a b i l i t i e s to sum to 1.  By the previous  separation r e s u l t s , then, x. = f . + m. X is is is s  and  x. = f . + m. X , f o r some optimal f- ,m. . isc is isc is is c  where X  g  i s the aggregate supply of the consumption good i n state s.  Hence, only two vectors,' ( f - | , . . .f S  I  s  )E^  and ( m ^ . .m^e'S . 1  s  are needed to span the space o f a l l agents optimal holdings of r i s k y and consumption assets. That i s , the date 1 market i s complete and y i e l d s aggregation and separation because o f the extended power u t i l i t y .  The  date 0 market i s incomplete and w i l l not aggregate, even with the extended power derived u t i l i t y .  However, the  normally d i s t r i b u t e d random variables a t date 0 induce a mean variance model which, o f course separates, even without a r i s k l e s s asset.  61. Chapter 4  Revelation of Individual  Endowments  Introduction In the l a s t c h a p t e r we examined the p o s s i b i l i t y o f p r i c e s r e v e a l i n g . preference data  ( a c t u a l l y an.aggregate preference  datum), which c o u l d be used by agents to make i n f e r e n c e s about f u t u r e prices.  H e r e , we study whether p r i c e s can s i g n a l i n f o r m a t i o n  the d i s t r i b u t i o n  o f endowments o f w e a l t h .  is necessarily multivariate,  Since t h i s  about  information  we s t u d y c o n d i t i o n s under which a  v e c t o r o f endowments can be s i g n a l l e d by p r i c e s .  In o r d e r  to  i n f e r the v a l u e o f a v e c t o r i n E u c l i d e a n n-space ( £ ) , one m u s t , a  i n g e n e r a l , be a b l e to observe a r e l a t e d v e c t o r i n &  m  where m ^ n .  o  For example, i f  ( x , y , z ) denotes a p o i n t i n £ , one can i n f e r  p o s i t i o n o f the p o i n t u s i n g t h r e e s p h e r i c a l c o o r d i n a t e s  the  (two  a n g l e s and a r a d i u s ) o r t h r e e c y l i n d r i c a l c o o r d i n a t e s (an a n g l e , a h e i g h t and a r a d i u s ) , but never w i t h two c o o r d i n a t e s o n l y . In the absence o f d e g e n e r a c i e s , i t i n o r d e r to convey n - d i m e n s i o n a l  i s n e c e s s a r y to have m >_ n  i n f o r m a t i o n w i t h an m - v e c t o r .  The  c o n d i t i o n m >_ n i s not s u f f i c i e n t and the problem must be s t u d i e d more c a r e f u l l y i n a s p e c i f i c s e t t i n g .  In the l a s t c h a p t e r , we  had m = n = 1, s i n c e p _ e ^ and G.e&^. In t h a t c a s e , a FRE e x i s t e d . 0  Most o f the r a t i o n a l  M  e x p e c t i o n s l i t e r a t u r e to t h i s date o n l y  d i s c u s s e s the e x i s t e n c e o f a FRE when n <_ 1.  F o r example, Radner  [1977] s t u d i e s the r e v e l a t i o n o f d i s c r e t e i n f o r m a t i o n p r i c e s , and Grossman [1976] has n=l parameter).  (an aggregate  (n=0)  by  information  To e s t a b l i s h the e x i s t e n c e o f a F R E , one must show t h a t a map from the parameter space (a s u b s e t o f & ) n  subset of a")  is invertible.  (If  to the p r i c e space (a  the s o c i a l endowment i s nonrandom,  we drop one dimension from the p r i c e s p a c e , by W a l r a s / ' l a w . )  It  g e n e r a l l y much h a r d e r to show c o n s t r u c t i v e l y t h a t a map from £  n  £  m  i s o n e - t o - o n e when m - v n  analyses of m u l t i v a r i a t e  > 1 than i t  i s when m = n = 1.  FRE's a r e by A l l e n [ 1 9 7 8 , 1 9 7 9 ] .  e s t a b l i s h e s the g e n e r i c i t y o f a FRE. A p r o p e r t y generic i f  arbitrarily  to  The She  ( o f an economy)  s m a l l adjustments to the parameters  endowments, p r o b a b i l i t i e s , e t c . )  is  is  (tastes,  do not d e s t r o y the p r o p e r t y , a n d ,  g i v e n an economy w i t h o u t the p r o p e r t y , t h e r e a r e a r b i t r a r i l y economies t h a t e x h i b i t t h a t p r o p e r t y .  close  Thus, a generic property  is  open and dense i n the f a m i l y o f economies under some a p p r o p r i a t e topology.^  The A l l e n r e s u l t s r e l y on the Whitney embedding theorem  of d i f f e r e n t i a l  t o p o l o g y ( s e e , e . g . , H i r s h [1976]) to get the  o f the e x i s t e n c e o f a FRE when m >_ 2n + 1.  genericity  Using a d i f f u s e n e s s  assumption on p r e f e r e n c e s and endowments, A l l e n [1979] weakens the d i m e n s i o n a l i t y requirement t o m > n + 1, i f  agents a r e not concerned  t h a t t h e p r i c e map i s many to one on a s e t o f p r o b a b i l i t y  zero.  results  requirement  are not r o b u s t w i t h r e s p e c t to r e l a x a t i o n o f the  The  t h a t m >_ n + 1, f o r J o r d a n and Radner [1977] p r o v i d e a g e n e r i c n o n e x i s t e n c e r e s u l t when m = n = 1.  The use o f d i f f e r e n t i a l  i s somewhat n o n c o n s t r u c t i v e and r e s u l t s u s i n g i t  topology  are hard to grasp  In o r d e r to be more c o n s t r u c t i v e , and have m=n, we s h a l l r e l y on the  better  known  implicit  intuitively.  largely  f u n c t i o n r e s u l t s which a r e based  on the rank o f the J a c o b i a n m a t r i x o f a t r a n s f o r m a t i o n .  These r e s u l t s  are o n l y o f a l o c a l n a t u r e , so we s h a l l s t u d y the e x i s t e n c e o f  l o c a l l y f u l l y informing r a t i o n a l expectations e q u i l i b r i a (LFRE) f o r which there e x i s t s some open set of information parameters such that economies r e s t r i c t e d to t h i s open set are FRE.  I t i s interesting  to note that the Jordan-Radner counterexample i s l o c a l l y f u l l y informing (LFRE) at a l l but one value of the exogenous parameters of the economy. The a n a l y s i s i s based on a power u t i l i t y economy (where agents have d i f f e r e n t powers, so aggregation f a i l s and d i s t r i b u t i o n of endowments matters). The market structure i s merely a complete market version of the three-date market used i n Chapter 3, although i t i s valuable to consider some r e l a t e d p a r a l l e l economies to get the  generic r e s u l t s .  Market s t r u c t u r e , notation and preferences As i n Chapter 3, there are three dates, with S states at date 1 and C states at date 2. date 0, n  s  T r a n s i t i o n p r o b a b i l i t i e s a r e , at  (s = 1, ..., S ) , and at date 1, s t a t e s, n  s c  (c = 1, ..., C).  At date 0, agent i has an endowment of y^ and trades to date 1 state s contingent wealth y^ .  At date 1, agent i trades to date 2  g  state c contingent wealth x.j > which he consumes, achieving u t i l i t y SC  (4.1)  V isc) x  =  Y  i  -1  x  Y i s c  i  (Ofri<l) or U ^ x ^ )  = log x  i s c  (y-0)  Exogeneous s o c i a l endowments of wealth at dates 0, 1, and 2 are, respectively Y  (4.2) (4.3)  =  is E_. X.  ISC  63.  Note t h a t , h e r e , Y and Y  g  a r e both g i v e n e x o g e n o u s l y , i n c o n t r a s t  to t h e models o f Chapter 3 , where Y = p prices.  The v a l u e s o f the Y  g  and Y  Q  g  = p  g  were endogenous  a r e exogenously r e v e a l e d t o a l l agents  a t date 0 , a l t h o u g h agents do  not know t h e v a l u e o f Y .  Markets  p r o v i d e a complete s e t o f f i n a n c i a l c l a i m s t o w e a l t h c o n t i n g e n t on the s t a t e o f n a t u r e a t t h e n e x t d a t e . q  =  s  Prices are:  date 0 p r i c e o f a c l a i m on $1 a t date 1 , c o n t i n g e n t on i n t e r m e d i a t e s t a t e s .  P  s c  =  date 1 p r i c e when i n t e r m e d i a t e s t a t e s o c c u r s o f a c l a i m on $1 a t date 2 , c o n t i n g e n t on f i n a l on f i n a l  state c.  I f agent i knows t h e p r i c e s ( p ^ > •••> p$rj> 9i> • • • » a t date 0 , he can s o l v e t h e usual dynamic programming problem f o r p o r t f o l i o demands.  Thus a t date 1, i n s t a t e s he performs  z n  max { x  s u b j e c t to <" - > 4  4  isc  c  c Psc i s c  E  U ^  sc  s  c  )  }  x  =  is  y  A t date 0 , he performs Vc  m a x  n  s sc n  u  i^ lsc x  )  {y > i s  subject to x  isc °Pti  (4.5)  l  m a  i n date 1 , s t a t e s and  E q y s  s  i s  =  y.  Denote t h i s market regime by M. tree i n Figure 4 . 1 .  It  i s r e p r e s e n t e d by t h e  (date 0 market)  sc (date 1 market)  Date Figure 4.3  Market Regime M"  I t i s also useful to consider two r e l a t e d market regimes. F i r s t consider the complete market M' f o r date 0 claims to money in date 2 state c, and f o r which there are no Let r  £  X-j  c  states.  E  date 0 p r i c e of claim to $1 at date 2, contingent on f i n a l state c.  =  payoff to i n d i v i d u a l i ' s p o r t f o l i o i f f i n a l state c p r e v a i l s a t date 2  = - i n n , the. unconditional  TT  intermediate  p r o b a b i l i t y of state  c  occurring,  The budget c o n s t r a i n t of agent i i s (4.6)  £ r x.  = y.  C C IC  J  l  Demands by i n d i v i d u a l i are determined by (4.6) and the f i r s t - o r d e r conditions (  4  . 7 ,  . u  'ld< 1d'  ^  .  "V'd  x  Equilibrium requires that demands s a t i s f y the market c l e a r i n g condition (4.8)  Z  i X i c  =  X  c  Now consider a more r e f i n e d complete market M" f o r claims to money i n f i n a l states i n which intermediate states e x i s t and the f i n a l state claims are also contingent on the i d e n t i t y of the intermediate s t a t e as well as the f i n a l s t a t e . Let p  =  s c  date 0 p r i c e of claim to $1 a t date 2, contingent on intermediate  state s a t date 1 and f i n a l state c a t  date 2. Agent 1 s- budget c o n s t r a i n t i s !  67. The f i r s t o r d e r c o n d i t i o n s f o r agent i  ic 1sc>  P s c / V s c ( s , t = 1 , . . . , S ; c,d = 1 , . . . , C )  'id  Ptd/Vtd  U,  u  are  (x  ( x  itd  )  The market c l e a r i n g c o n d i t i o n i s  (4.3).  Observe t h a t the e q u i l i b r i u m t h a t r e s u l t s i n markets NT a l s o s a t i s f i e s the e q u i l i b r i u m c o n d i t i o n s f o r market M " , where x  ( - ) 4  isc  P  9  =  x  ic  (Vsc/\  = s c  )  r  c  Markets M and M" p r e s e n t the same o p p o r t u n i t y 1  and so market v a l u e s must be the same i n b o t h .  • s e t to a l l  individuals  Assume M" has a unique  2) equilibrium.  '  The r e a s o n f o r i n t r o d u c i n g markets M  1  and M" as a s t e p i n  the  a n a l y s i s o f t h e market M i s t h a t Arrow [1953] has shown t h a t , i f  all  i n d i v i d u a l s a r e informed o f ex p o s t e q u i l i b r i u m p r i c e s i n market M, individuals will final  f a c e the same o p p o r t u n i t y  s t a t e p a y o f f s i n M and i n M " .  s e t s and a c h i e v e the same  S i n c e M' and M" a r e e q u i v a l e n t ,  we can a n a l y z e M through c o n s i d e r a t i o n o f the s i m p l e r , market M ' . Arrow has a l s o shown t h a t e q u i l i b r i u m p r i c e s i n M and M" a r e r e l a t e d by ^s sc ~ sc * p e r i o d budget c o n s t r a i n t s i n M and a p p l y i n g p  Summing i n d i v i d u a l  final  the market c l e a r i n g r e l a t i o n s  p  (4.2)  and ( 4 . 3 )  E„p JL = Y c sc c s c  c  yields  6 8 .  T h e r e f o r e , e q u i l i b r i u m p r i c e s i n M and M' are r e l a t e d by  '  (4  % - K/'i^c K c ' V ' A  ,0)  P  (4.11)  The r e v e l a t i o n o f  = (Vsc c / ;  sc  ) ( r  c s>  •  / q  information  Assume t h a t a l l agents know t h a t everyone has power o r l o g u t i l i t y o f the form ( 4 . 1 ) , and they a l l u t i l i t y exponents from the s e t  know t h a t a l l agents have Yj>-  However, agent i does  not know how much w e a l t h y . the o t h e r agents ( j = l , . , .  ' jfi)  have.  There may be a c l a s s o f s e v e r a l agents w i t h the same y^ , i n which case the aggregate demand f u n c t i o n f o r the c l a s s w i l l as i f  be the same  a l l w e a l t h o f the c l a s s were bestowed on a s i n g l e agent w i t h  the same u t i l i t y f u n c t i o n as a l l members o f the c l a s s .  Thus,  a l t h o u g h an agent knows h i s p e r s o n a l w e a l t h , i n g e n e r a l , he w i l l not know the aggregate w e a l t h y . o f h i s c l a s s , so t h a t a l l  agents I  d e s i r e to i n f e r the whole v e c t o r o f w e a l t h s (y-j,  ...,yj)  3)  e £ .  /  A t date 0 , agents do not know the y^ o r the aggregate Y , but do know the aggregate s o c i a l endowments o f date 1 w e a l t h and d a t e 2 w e a l t h {X > .,  (y.)  s  As i n Chapter 3 , they have homogeneous  c  probability  {Y >  beliefs {^s^sc^*  T n e y  d e s i r e to i n f e r the v a l u e o f  and hence the s t a t e c o n d i t i o n a l date 1 p r i c e . v e c t o r  (P  It  this  i s c o n v e n i e n t to c o n s i d e r f i r s t the market M' i n  s e t t i n g and then g e t q  s  and p  s c  s c  )-  i n M by u s i n g ( 4 . 1 0 ) and ( 4 . 1 1 ) .  Agent i ' s demand f u n c t i o n a t date 0 i n M' can be d e r i v e d as -i 1 - i f il where -= (y i) d d d ic t  I  6  x  =  J  E  ( r  V  1  6  / l T  }  r  Differentiating  totally  with respect to x. , y  i n d i v i d u a l s and n o t i n g t h a t X  i  and r , Q  summing o v e r  i s f i x e d , so t h a t E-jdx.^ = d X = 0 ,  £  G  y i e l d s the f o l l o w i n g r e l a t i o n expressed i n matrix  form:  (A - X'GBX) d r + X ' G dy = 0 where A = diag X  S  [x  1 c  [ l ^ V " ^ ]  B = diag  [^+1]  G = diag  [y. ] _1  dr = ( d r . , , . . . ,dr(0,' dy = (dy-,  and " d i a g " denotes a d i a g o n a l  ,.v.,dy )' I  matrix.  Assuming p r i c e s are d i f f e r e n t i a t e (4.12) A l s o , (4.10)  functions  d r = - (A - X ' G B X ) "  1  X ' G dy  o f w e a l t h endowments, .  can be e x p r e s s e d as  (4.13)  q = EnF r  where q = (q ,.. . ,q^) 1  E = diag n =  [ir  s c  F = diag r =  1  :  [TT /Y ] S  S  ]  [ X ^ ]  (r^,...,)  Hence, changes i n observed p r i c e s a r e r e l a t e d to changes i n w e a l t h endowments by (4.14)  dq = Hdy  where dq = ( d q  dq )'  1  s  H =-EnF(A-X GBX)~ X G ,  A p p l y i n g the i m p l i c i t of ( q - | , . . . , q ) s  ,  f u n c t i o n theorem, (y..) i s l o c a l l y a f u n c t i o n  i f t h e c o e f f i c i e n t m a t r i x H has rank I.  i s t h e c a s e , then p p  1  s c  i s a function of r  If  i s l o c a l l y a f u n c t i o n o f (q-| and q  by ( 4 . 1 1 ) and r  this  q^) because  i s a function of  ( y - ] > ' " > y j ) by t h e uniqueness o f p r i c e s i n economy M ' .  Hence, a  LFRE would e x i s t . Observe t h a t EnF i s S x C , ( A - X ' G B X ) C l e a r l y , rank (H) < I i f C < I o r S < I, communicate a l l o f the I - d i m e n s i o n a l  i s CXC, and X ' G i s C x i .  - 1  so t h a t p r i c e s q cannot  -(y.) i n f o r m a t i o n .  t h i s need n o t r u l e out t h e e x i s t e n c e o f a FRE. C >_ I and S _> I, f o r rank (H) = I. Case 1:  However,  M o r e o v e r , even i f  i t i s s u r p r i s i n g l y hard t o e s t a b l i s h c o n d i t i o n s The problem r e q u i r e s an a n a l y s i s by c a s e s .  C <_ S .  Theorem 4 . 1 : A s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e o f a FRE ( g l o b a l , n o t j u s t l o c a l ) i s t h a t rank (n) = C . Proof:  S i n c e E and F a r e n o n s i n g u l a r d i a g o n a l m a t r i c e s , rank (EnF)  rank (n) = C .  Thus t h e mapping ( 4 . 1 3 ) o f r to q i s  invertible.  By o b s e r v i n g q , an agent may i n f e r r and hence { p }  by ( 4 . 1 1 ) .  c r  T h i s a l l o w s him t o s e l e c t h i s date 0 p o r t f o l i o were f u l l y  i n M as though he  informed. Q.E.D.  Note t h a t i n Theorem 4.1 agents do n o t l e a r n a l l about y = ( y - j . . , y j ) , but j u s t enough about i t r and { p } c r  o b t a i n s ) t o behave o p t i m a l l y .  information  ( i . e . , about which  Case 2 :  S < min ( I , C ) .  With I > S and C > S  not enough dimensions o f v a r i a t i o n i n (q-, p a r t i c i p a n t s o f the v a l u e o f e i t h e r  t h e r e are q^) to i n f o r m market  (r.,,..'. .r^)  or (y-|,. • • , y ) . s  Thus, we h a v e : Theorem 4 . 2 :  If  S < min ( I , C )  n e i t h e r a LFRE nor FRE e x i s t ,  u n l e s s the market i s degenerate to a l l o w a g g r e g a t i o n o f (thus e f f e c t i v e l y Case 3 :  r e d u c i n g I)  I <_ S < C .  or of f i n a l  individuals  s t a t e s (thus r e d u c i n g C ) .  T h i s case i s p r o b a b l y the most i n t e r e s t i n g and  c e r t a i n l y the most i n t r a c t a b l e .  The i n t e r e s t stems from the  fact  t h a t t h i s case p l a c e s no upper bound on the number o f p o s s i b l e s t a t e s o f the w o r l d .  There are SC p r i c e s p  g c  which an agent  would l i k e to know i n o r d e r to s o l v e h i s dynamic programming problem i n market M a t date 0 . and {ir },  an i n d i v i d u a l  By knowing the p r o b a b i l i t i e s  can reduce t h i s  information  i^ } sc  collection  p r o b l e m , by u s i n g ( 4 . 1 0 ) and ( 4 . 1 1 ) , to one o f f i n d i n g o u t C dimensions o f i n f o r m a t i o n  f o r the r ' s c  to supplement the . i n f o r m a t i o n  ( q - | , . . . , q ) which he can o b s e r v e i n the date 0 m a r k e t . s  if  C > S , he s t i l l  However,  cannot l e a r n a l l o f the i n f o r m a t i o n he needs  u n l e s s i t a c t u a l l y has a l o w e r dimension than C.  S i n c e the r  are determined by o n l y w e a l t h l e v e l s , y ^ , t h e r e a r e FRE's arbitrarily  l a r g e numbers, C, o f p o s s i b l e f i n a l  as the number o f c a t e g o r i e s o f a g e n t s , I, than S , the number o f o b s e r v a b l e s i g n a l s Unfortunately i t results in this case.  's  for  s t a t e s as l o n g  does not become l a r g e r (prices).  i s very hard to get p o s i t i v e o r n e g a t i v e F i r s t , here i s a somewhat n e g a t i v e  result.  Counterexample:  If  I <_ S < C , t h e r e a r e economies f o r which  H = -_ E n F ( A - X G B X ) " X - ' G has rank l e s s than I. ,  F o r example, l e t  1  S = I = 2 , C = 3 , and l e t n-, A-0.-. and, l e t n.-, + ti may t h i n k o f (m>n2  a s . a v e c t o r o f • p r o p o r t i o n s - o f weal t h r e a l l o -  c a t i o n s dy = ( d y , d y ) ' 1  (dY=0). is  Let e =  = 0'.. One .  2  2  such t h a t t o t a l weal th  (5 ,c ,? ) 1  2  1  3  Y  y ^ y  =  = - F(A-X'GBX)" X'G(n ,n ) . 1  1  1  2  is  constant  It  shown i n t h e Appendix to t h i s c h a p t e r t h a t we can s e l e c t n  of f u l l  rank such t h a t i t i s a p r o p e r c o n d i t i o n a l  m a t r i x and M The  probability  = 0 (and hence rank H <. 1 < I ) .  counterexample shows t h a t i t i s n o t always the case  t h a t t h e s t a n d a r d hypotheses o f t h e i m p l i c i t f u n c t i o n hold is  2  and hence guarantee t h e e x i s t e n c e o f a LFRE.  theorem  It  not n e c e s s a r i l y a counterexample to t h e e x i s t e n c e o f a LFRE.  There a r e many 1-1 mappings which have a s i n g u l a r J a c o b i a n a t some p o i n t i n t h e i r domain. f'(0)  For example, l e t f ( x ) = x .  Since  = 0 , t h e J a c o b i a n i s s i n g u l a r a t 0 , even though the f u n c t i o n  i s . s t r i c t l y monotone i n c r e a s i n g and i n v e r t i b l e . (and has n o t been found) i s a d i f f e r e n t i a t e  What i s needed  path P g i v e n by  y(t)  = (y (t),...,y (t))  q(t)  - q(0) = - / * E n F ( A - X G B X ) " X G d y ( x ) = 0 ( t e [ 0 , l ] )  1  I  f o r t e [ 0 , l ] such t h a t ,  1  ,  = 0  but r ( t ) 2  - r(^)  = J*|  t  ( A - X ' G B X J - ^ ' G dy(x) f 0  f o r almost a l l t-, < t 2  There would be no L F R E ' s anywhere a l o n g t h i s path s i n c e q would be c o n s t a n t a l o n g the p a t h , even though r c o n t i n u o u s l y varies.  An  individual  cannot t e l l  where the economy i s on t h e  p a t h , but needs the i n f o r m a t i o n f o r h i s dynamic program.  There a r e two p o s i t i v e r e s u l t s on the e x i s t e n c e o f however.  For e x a m p l e , they do e x i s t .  Theorem 4 . 3 : has  If,  i n economy M ' , the m a t r i x o f o p t i m a l  holdings X  rank I and I <_ S <_ C, and agent 1 has -1 < Y-| 1 0 ,  exist  LFRE's,  there  a r b i t r a r i l y - s m a l l - c h a n g e s i n {T^}, { T T } , { X } and y SC  c  t h a t the r e s u l t i n g economy has a LFRE ( i . e . , where q i s an i n v e r t i b l e is  function of y ) .  changes i n t h e exogenous  Comments:  remains a LFRE under a r b i t r a r i l y parameters o f the economy.  function s u f f i c i e n t  LFRE i s an open  rank X m a t r i c e s .  a power u t i l i t y economy M' has rank (H) =  (and hence i s a LFRE), i t  the i m p l i c i t  locally  That i s , the s e t o f LFRE economies  dense i n the s e t o f economies o f f u l l Moreover, i f  such  1  I,  small  That i s ,  condition f o r existence of a  property.  Coupled w i t h t h e c o u n t e r e x a m p l e , t h i s theorem s u g g e s t s  t h a t the m a t r i x probabilities  [ I ^ T T ^ ] o f j o i n t i n t e r m e d i a t e and f i n a l  is crucial  i n d e t e r m i n i n g whether enough  state information  about market M' p r i c e s r can be communicated by date 0 market M prices q.  This matrix of p r o b a b i l i t i e s c o n s t i t u t e a v e i l  that  agents may o r may not be a b l e to see through a t d a t e 0 . The assumption about the rank o f X w i l l Theorem 4 . 4 .  be weakened i n  Proof o f Theorem 4.3 F i r s t , we s h a l l show that we can perturb the IT'S to ensure rank (H) = I , so that a LFRE e i x s t s by the i m p l i c i t function theorem (see, e.g.,  Dieudonne [1960], Rudin [1964]), provided that A-X'GBX i s  invertible.  Recall that H = - EnF(A-X'GBX)" X'G 1  = -  ir^ j  where  diag[Y  _ 1 s  ][rr | ]D s  c  Pr (intermediate s t a t e s | f i n a l s t a t e c)  = — — — =  c D = diag[X ](A-X'GBX)" X'G  and  1  c  C l e a r l y , we can vary the  TT  and T T  s  S C  SO  that the  T T |c s  change but the  and hence X, A, G, B and D are constant, since they are formed in economy M ' , which i s unchanged.  The rank o f D i s I since the  rank o f X i s I, and D i s j u s t X pre- and p o s t - m u l t i p l i e d by i n v e r t i b l e matrices.  Let D be the square submatrix formed by taking the f i r s t  I rows o f D and suppose w.l.o.g. that rank (D) = I. delete rows of [  7T S  | ] so that we can assume C  A l s o , we may  w.l.o.g. that S = I.  Then, the determinant o f [TT I ]D i s a multinomial  i n (TT I  Sj C  S = 1,  Sj c  I - 1; c = 1, ..., C>, say  det(Dr | ]D) = m(ir s , c ) s  c  .  I f we cannot s l i g h t l y perturb the ^ ^ so that rank (H) = I , then ( | ) 0 °P t ( i n the product o f projections o f m  7 r  E  s  o n  s o m e  en  s  e  c  S-l S - simplices into &  ). As i n the discussion o f the counterexample  i n the Appendix, t h i s can only occur i f a l l the c o e f f i c i e n t s o f m are i d e n t i c a l l y 0.  But, t h i s cannot be so, f o r i f we define r s  l  c  1 i f s = c,  1 0 otherwise  then m ( T r | ) = det(D) f 0 . s  Hence, f o r some a r b i t r a r i l y  c  small per-  t u r b a t i o n o f the TT i ' s , H has rank I and a LFRE e x i s t s . sIc To see t h a t the p r o p e r t y  "rank (H) = I"  o c c u r s on an open s e t ,  note t h a t p r i c e s , demands, e t c . a r e a l l c o n t i n u o u s i n the exogenous parameters o f the economy ( i f  p r i c e s are d i f f e r e n t i a b l e )  the s e t o f SXI m a t r i c e s w i t h f u l l nonzero d e t e r m i n a n t ) Now i t  i s open i n £  rank ( i . e . , SI  and t h a t  having s u b m a t r i c e s w i t h  .  o n l y remains to be shown t h a t the i n v e r t i b i l i t y  (A-X'GBX) i s an open and dense p r o p e r t y i t s e l f .  of  Clearly, if  A-X'GBX  i s n o n s i n g u l a r , the b e h a v i o r o f the economy i s s u f f i c i e n t l y  continuous  f o r the n o n s i n g u l a r i t y to h o l d under s m a l l p e r t u r b a t i o n s o f  the  parameters o f the economy. If  -1 < . y  < 0 ( i = 1 , . . . . , I ) , , the m a t r i x - A + X'GBX has a l l  p o s i t i v e elements and a dominant d i a g o n a l ( c f . and hence i s p o s i t i v e d e f i n i t e and i n v e r t i b l e . a dominant d i a g o n a l i f l cc m  I c a  >  t h e r e e x i s t <* * l cdl d= I dfc m  a  Gale-Nikaido  •  [ 1 9 6 5 ] )  A CxC m a t r i x M has  > 0 (c = 1 , . . . , Q ) such t h a t  d  (c = 1 . . . . .  C).  .  In t h i s c a s e , t a k e a = r . c c Suppose t h a t - 1 < Y-J ±0,  even though o t h e r a g e n t s ' exponents  may not s a t i s f y t h i s i n e q u a l i t y . the s o c i a l endowments {X > c  it  By i n c r e a s i n g the w e a l t h y-j and  i s p o s s i b l e to move agent 1 a l o n g h i s  Engel curve (a ray through the o r i g i n ) w i t h o u t and o t h e r a g e n t s ' a l l o c a t i o n s .  This occurs i f  varying prices dX  L  =  9X  {r > c  l_c dy, = l c  _____ dy j  x  ayi  |  y  }  n  T h i s has the e f f e c t o f a d d i n g more and more o f a dominant d i a g o n a l m a t r i x to - A + X'GBX.  A t some p o i n t i n the p r o c e s s , the  matrix  -A + X'GBX i t s e l f must assume a dominant d i a g o n a l and become invertible.  M o r e o v e r , the m a t r i x -A+X'GBX i s l i n e a r i n y-j,  p r o v i d e d the X ' s v a r y as above so t h a t x ^ c  Hence d e t (-A + X'GBX)  is a C  does not v a n i s h e v e r y w h e r e . t h e r e must be a r b i t r a r i l y  c  i s l i n e a r i n y-j.  degree p o l y n o m i a l i n y^ which  S i n c e i t s z e r o e s are i s o l a t e d ,  small p e r t u r b a t i o n s o f y-j and ( X )  that  c  make A - X ' G B X i n v e r t i b l e .  Q.E.D.  The next theorem weakens the assumption about the rank o f X . Theorem 4 . 4 :  With the same hypotheses as i n Theorem 4 . 3 , e x c e p t  t h a t the rank o f X may be l e s s than I,  there e x i s t a r b i t r a r i l y  small  changes i n p r o b a b i l i t y b e l i e f s {TT } , {TT } and endowments { y . } , S  such t h a t a LFRE e x i s t s .  SC  1  M o r e o v e r , the economy w i l l  {X„} c  be p e r t u r b e d  to a neighbourhood o f i t s exogenous parameters on which rank (X) = J f o r some c o n s t a n t J <_ I, Comment:  and on which L F R E ' s  exist.  The p r o o f o f t h i s r e s u l t uses the rank theorem (Dieudonne  [1960, p. 2 7 3 ] , Rudin [1964, p. 1 9 8 ] ) , which i s a g e n e r a l i z a t i o n o f the i m p l i c i t  f u n c t i o n theorem to mappings which are o f l o c a l l y c o n -  s t a n t (but not n e c e s s a r i l y f u l l )  rank.  The rank theorem e s t a b l i s h e s  the e x i s t e n c e o f a mapping y -> ze£^ such t h a t r function of z .  i s an  invertible  The t e c h n i q u e s o f Theorem 4 . 3 . c a n then be used to  a d j u s t the j o i n t p r o b a b i l i t i e s the v a l u e o f z and hence r .  {  Tr  1T s  sc  } so t h a t p r i c e s q communicate  The i m p o r t a n t p o i n t here i s t h a t i t may  not be n e c e s s a r y to s i g n a l a l l an adequate summary s t a t i s t i c .  the endowment i n f o r m a t i o n ,  since z is  Rank Theorem:  I c Let y° e N c R , where N i s an open s e t , and f : N -> &  be a continuously  d i f f e r e n t i a t e mapping such t h a t , i n N, the rank  of the Jacobian matrix i s a constant J . Then there e x i s t : 1.  an open neighbourhood U £ N of y° and a function ,1 onto  3  •1  such that g and g~ are continuously different!'able (-1.D  1  = {y  G  1} i s the open unit ball  y | <1, i = 1  &  1  (here,  i  in &*), and 2.  an open neighbourhood V P f(U) of f(y°) and a function f h : (-1,1) '  such that f = h f 0  0 0  g  1 1 -^4—V onto  with h and h  1  continuously d i f f e r e n t i a b l e , J  where  f ° : (-1.D -> (-1,D 1  H,i)  C  by f ° (  J  f°  Zj) = ( z  Z]  -(-1,1)  Z j  r  , 0,  1  This theorem says t h a t , i f the mapping f has constant rank J on some open s e t , then the action o f f can be summarized by J variables (z-j,  Z j ) . I f J = I , i t i s the usual i m p l i c i t func-  t i o n theorem. Proof of Theorem 4.4 Theorem 4.4 i s proved by f i r s t perturbing y-j and ( X ) to a point c  where A-X'GBX has f u l l rank, and then perturbing y to some point about  0)  which there i s a neighbourhood on which the p r i c e map f : y —*~--re& has a Jacobian of constant rank J while A-X'GBX stays of f u l l rank as i n the proof of Theorem 4.3.  The desired r e s u l t follows by  varying TT | as i n the proof of Theorem 4.3 to ensure that the s  c  prices q are a 1-1 function of the variables z-,, ..., Z j whose existence i s established by the Rank Theorem. F i r s t , perturb y-, and ( X ) to a neighbourhood on which c  A-X'GBX i s i n v e r t i b l e .  Then note that y can be perturbed to a  point y° about which there i s an open neighbourhood N on which rank (X) i s l o c a l l y a constant, say J . This occurs because {y : rank (X) ii J } i s open f o r any J ' , due to the c o n t i n u i t y of 1  the mapping y —> det (X) i n market M  1  submatrix of X.  where X i s any square  Since rank (X) i s bounded above by I , we can  choose any y° f o r which the corresponding matrix X° has rank J where J i s the l i m sup  of the ranks o f the X matrices as the  endowment vector approaches y. Let f : y —> r be the mapping sending endowments to prices r in market M'.  The Jacobian matrix —  = - (A-X'GBXrVG  3y  has constant rank J i n the neighbourhood N, since i t i s merely the matrix X  1  pre- and p o s t - m u l t i p l i e d by i n v e r t i b l e matrices.  Now, apply the Rank Theorem. (z.,, r = h(z-|,  Z j , 0,  Using the notation  0) = f° g(y), one can see that prices 0  Z j , 0, ..., 0) = R ( z ) , say, where z = (z-,  Zj).  By  (4.13) q = E Fnr = diag[Y  where the c o n d i t i o n a l  ]  - 1 s  C - | l d i a g [X ] s  probabilities  c  TT | s  c  0  R (z)  a r e d e f i n e d i n the  c  proof  o f Theorem 4 . 3 .  .'.  M  =  d i a g [Y - ] 1  3h/3z  diag [ X l  has f u l l  — -ai  1  3z  By c o n s t r u c t i o n ,  [V . ]  rank J , so i t  i s p o s s i b l e to use  the same type o f argument as i n t h e p r o o f o f Theorem 4 . 3 to an a r b i t r a r i l y full  rank J .  s m a l l p e r t u r b a t i o n o f { T | } SO t h a t S  C  That i s , q i s l o c a l l y an i n v e r t i b l e  find  a l s o has  3q/3z  function of  z.  Thus, knowing the q ' s , agents can i n f e r the z ' s , and from the z ' s they can i n f e r p r i c e s r = h ( z ) , which a l l o w s them to s e l e c t optimal  demand f u n c t i o n s f o r {y.  their  } a t date 0 .  is Note t h a t agents cannot l e a r n a l l o f t h e y ' s . know the r e l e v a n t i n f o r m a t i o n may n o t have any n a t u r a l  impounded i n z .  interpretation  They need o n l y  Unfortunately,  z  as an o b s e r v a b l e economic  variable. ^  Q.E.D.  4  Conclusion In t h i s c h a p t e r , we have s t u d i e d whether o r not t h e r e locally fully  informing rational  expectations e q u i l i b r i a  exist  (LFRE)  in  a power u t i l i t y economy where agents a r e u n c e r t a i n about f u t u r e p r i c e s because they a r e u n c e r t a i n about the c u r r e n t d i s t r i b u t i o n wealth.  It  of  was shown t h a t L F R E ' s " u s u a l l y o c c u r " i n the. g e n e r i c  sense t h a t they o c c u r on an open and dense s e t o f  parameter!*zations  o f power u t i l i t y economies i n which a t l e a s t one agent has a m i l d  r e s t r i c t i o n on h i s u t i l i t y exponent, provided that there are more states of the world (and hence markets) at date 0 than there are individuals.  Prices may not completely  reveal endowments, but  they tend to reveal enough information to forecast future p r i c e s . I f there are not more markets than people, then a FRE e x i s t s when there are more date 1 states than date 2 states of the world. For a l l of these r e s u l t s , a key issue i s whether or not the structure of t r a n s i t i o n p r o b a b i l i t i e s from date 1 states to date 2 states i s r i c h enough to signal information about date 1 prices with date 0 p r i c e s . Note that although the analysis of t h i s model used the complete market structures M' and M",  so that the S markets at date 0  i n M w i l l , by Arrow's theorem, allow ultimate a l l o c a t i o n s as i f the markets were complete i n date 2 goods, the H i r s h l e i f e r - M a r s h a l l r e s u l t does not y i e l d the r o l l o v e r strategy that flawed the l a s t chapter.  This happens because agents must know date 1 r e l a t i v e  prices to optimally convert the S s e c u r i t i e s of the f i r s t period to the C s e c u r i t i e s of the second period.  The next chapter deals with  -constructive models that are not flawed by the r o l l o v e r strategy.  Footnotes to Chapter 4. 1.  Openness alone i s not a strong c o n d i t i o n , since the u n i t b a l l  is open i n & ,  but i n many senses i s an i n s i g n i f i c a n t part of g. .  n  n  Density alone i s not strong since the property may perturbations.  f a i l under small  A l t e r n a t i v e l y , some w r i t e r s connote g e n e r i c i t y with  f u l l Lebesgue measure.  Even i f the underlying topology i s the usual  metric topology on £ , neither notion o f g e n e r i c i t y implies the other. n  Although sets of f u l l Lebesgue measure are dense, they are not n e c e s s a r i l y open.  On the other hand, there are open and dense sets  of l e s s than f u l l Lebesgue measure.  As an example, consider a modi-  f i c a t i o n of the c o n s t r u c t i o n of the Cantor s e t obtained by d e l e t i n g shorter and shorter closed i n t e r v a l s from the u n i t i n t e r v a l i n  so  that i n the l i m i t the remainder (a modified Cantor set) has measure (rather than zero, as f o r the Cantor s e t ) .  This set i s closed and  no-  where dense, so that i t s complement i s open and dense, but of l e s s than f u l l measure, • c f . , 2.  I f Y I = Y2  =  Taylor [1973, p. 94]. =  Yj»  by footnote 2, Chapter 3.  preferences aggregate and prices are unique Hence, prices should also be unique i f the Y  do not vary "too much" c r o s s - s e c t i o n a l l y .  I f prices are not unique, we  must assume a l l agents know which r u l e the Walrasian auctioneer follows in s e l e c t i n g p r i c e s . 3.  A l t e r n a t i v e l y , one may  think of one agent per c l a s s , so that agent  desires to know only 1-1 other endowments, given that s o c i a l endowments are true s o c i a l endowments l e s s h i s demands.  E s s e n t i a l l y , the same re-  s u l t s of t h i s chapter hold under these circumstances with obvious modif i c a t i o n s , such as replacing I with  1-1.  4.  A reasonable conjecture i s that, i f a l l agents or classes of  agents have d i f f e r e n t generically.  power u t i l i t y exponents, then J = I holds  That i s , the set of endowments and p r o b a b i l i t i e s  which rank (X') = I is dense and open in the set of a l l and p r o b a b i l i t i e s .  The conjecture i s true for 1 = 2 , r  agent i trades to x^ e & . some ae £ . origin.  Then rank (X ) < I i f f  That i s , x-j and x  1  2  endowments  f o r , suppose  x^ = a x  2  for  l i e on the same ray through the  Since the Engel curves are rays through the o r i g i n with  power u t i l i t y that d i f f e r only when the powers d i f f e r , that y-j  for  =  Y  a n c 2  ' ^l  =  a  ^2'  i t must be  83.  Appendix to C h a p t e r 4 D e t a i l s o f t h e counterexample p r o v i d i n g a J a c o b i a n o f l e s s than  full  rank We d e s i r e ir „ > 0 such t h a t sc — (4A.1)  ^1^11  +  (4A.2)  7T  2 21 1T  ^1^12  +  (4A.3)  1T  2 22 1T  ^I^IS  (4A.4)  t- ^ }  u  + C TT 2  + 5 TT 3  1 2  +  ^1  =  ^2  =  ^3 = 0  13  (4A.5)  5-j TT-, 2  where we a r e g i v e n {ir }» U ) and U > c  ^2^23  =  s  + Recall  c  +  ^3^23  =  °*  the i n t e r p r e t a t i o n  that  n - = d y , so t h a t n  i  dr =-(A-X'G B X ) X ' G d y _ 1  =  E,  Equations (4A.1) -  and  F dr  ( 4 A . 3 ) must be s a t i s f i e d f o r the TT  sc  to be c o n d i t i o n a l  p r o b a b i l i t i e s and e q u a t i o n s ( 4 A . 4 ) and (4A.5) say t h a t nFdr = 0 hence dq = EnFdr = 0 .  E  and  Now,  s,c V s c c 5  =  ^.c^s'scV'c^fc  = E X dr = z.dy^  (by d e f i n i t i o n  o f ir )  (by W a l r a s ' law o r ( 4 . 6 )  and ( 4 . 8 ) )  = 0 Hence e q u a t i o n s  (4A.1) -  (4A.5) are r e d u n d a n t , so we may drop  l e a v i n g 4 e q u a t i o n s i n 6 unknowns.  Solving for  the o t h e r  (4A.5),  variables  i n terms o f TT^ and TT-^ one can see t h a t the f e a s i b l e r e g i o n  is  r e p r e s e n t e d by  C4A.6) (4A.7)  0  (4A.8)  1 2  £_  T  °£ lSl ll " 7r  assuming w . l . o . g . a r e not a l l S i n c e TT  <_ T r i T r  15  that £  3  >0.  7r  7 r  2  l ? 2 1 2 ± $3^3 ' 7 r  S i n c e -z-  o f the same s i g n , so assume w . l . o . g .  TT and £ 2  3  TT a r e a l l 3  that ^  < 0.  p o s i t i v e , t h e r e i s a nonempty open 2  s u b s e t N o f the p o s i t i v e q u a d r a n t . o f & , h a v i n g the o r i g i n as a l i m i t point,  which i s c o n t a i n e d i n the f e a s i b l e s e t f o r  g i v e n by ( 4 A . 6 ) and ( 4 A . 8 ) .  Now, the d e t e r m i n a n t  two columns o f n i s a m u l t i n o m i n a l f o r the o t h e r  TT 'S SC  Where i n N o n l y i f i n N, a l l  its  coefficients.)  in i r  n  and TT  12  (7r ,ir ) 1 1  1 2  o f the  first  (after  solving  i n terms o f t h e s e ) , and hence can v a n i s h e v e r y it  v a n i s h e s everywhere i n & .  (If  it  vanishes  d e r i v a t i v e s must a l s o v a n i s h , and hence so must But s e t t i n g -ni-nn = -Hi and i r  7T ) g i v e s a nonzero d e t e r m i n a n t 2  Hence one can choose n o f f u l l  12  the  = 0 (so t h a t TT TT 2  o f the f i r s t two columns o f n. rank to s a t i s f y  (4A.1)  to  (4A.5).  22  Chapter 5  P r i c e s R e v e a l i n g Aggregate R i s k A v e r s i o n , Impatience and Probability Beliefs  Introduction The two p r e v i o u s c h a p t e r s  showed how p r i c e s can r e v e a l  a g g r e g a t e p r e f e r e n c e and endowment i n f o r m a t i o n economy. beliefs  The o t h e r major f a c t o r  i n an exchange  influencing prices is  (when a s s e t s a r e r i s k y ) .  T h i s c h a p t e r s t u d i e s the  o f when p r i c e s r e v e a l p r e f e r e n c e s and p r o b a b i l i t y types o f models are d i s c u s s e d .  probability question  beliefs.  Two  One i n v o l v e s a one p e r i o d w o r l d  w i t h consumption a t two dates and t h r e e s e c u r i t i e s ; c u r r e n t  con-  s u m p t i o n , a r i s k l e s s bond and a c l a i m on r i s k y f u t u r e c o n s u m p t i o n . The r e s u l t i n g two r e l a t i v e and p r o b a b i l i t y  p r i c e s may r e v e a l aggregate r i s k  b e l i e f parameters.  p e r i o d s , so t h a t the i n t r o d u c t i o n market c r e a t e s t h r e e r e l a t i v e  The o t h e r model i n v o l v e s o f a term s t r u c t u r e  p r i c e s t h a t may r e v e a l an i m p a t i e n c e  s h o u l d be i m p o r t a n t  o f these p a r a m e t e r s . vation for interest •mat-ton  involve  b e l i e f s o n l y , and the two p r e v i o u s  c h a p t e r s i n v o l v e the r e v e l a t i o n o f p r e f e r e n c e s , so i t to ask why i t  parameters.  e x p e c t a t i o n s models i n the l i t e r a t u r e  the r e v e l a t i o n o f p r o b a b i l i t y  two  i n the bond  p a r a m e t e r , i n a d d i t i o n to r i s k a v e r s i o n and p r o b a b i l i t y Typical rational  aversion  i s reasonable  to a n a l y s e the j o i n t  P a r t o f the r a t i o n a l e  revelation  i s r e l a t e d to the m o t i -  in r e v e l a t i o n of preferences alone.  Agents'  info  about .aggregate p r e f e r e n c e s may b e b a s e d on e c o n o m e t r i c  o b s e r v a t i o n o f p r i o r markets  :  (as i n , f o r e x a m p l e , F r i e n d and Blume  [ 1 9 7 7 ] ) , c a s u a l i n f e r e n c e s drawn from o b s e r v a t i o n o f p r i o r  markets,  a n d / o r i n f e r e n c e s drawn from o b s e r v a t i o n o f c u r r e n t market p r i c e s .  Only the l a s t method r e q u i r e s r a t i o n a l t h a t developed i n Chapter 3 , and i t  e x p e c t a t i o n s machinery  i s necessary only i f  preferences  a r e u n s t a b l e and cannot be a c c u r a t e l y f o r e c a s t from p r i o r The f i n a n c e l i t e r a t u r e has never s e r i o u s l y s t u d i e d the  like  data.  possibility  o f u n s t a b l e p r e f e r e n c e s , presumably because r e s e a r c h e r s have tended to a s c r i b e v a r i a t i o n s  i n p r i c e s o f r i s k y a s s e t s to changes i n b e -  l i e f s o r i n f o r m a t i o n , but not to changes i n p r e f e r e n c e s . specifying functional  Without  forms f o r u t i l i t i e s and s e c u r i t y p a y o f f  dis-  tributions,  it  information  and changes i n p r e f e r e n c e s as s u p e r i o r e x p l a n a t i o n s  variations  i s g e n e r a l l y hard to d i s t i n g u i s h between changes i n  in security prices.  As a r e s u l t ,  for  the h i s t o r y o f a s s e t  p r i c e s i s u s u a l l y e x p l a i n e d by a sequence o f i n f o r m a t i o n  arrivals,  because t h i s i s more amenable to e m p i r i c a l a n a l y s i s than a sequence o f changes i n p r e f e r e n c e s . However, t h e r e i s some anomalous e v i d e n c e about s e c u r i t y  prices  t h a t i s hard to t o t a l l y a s c r i b e to a sequence o f i n f o r m a t i o n  arrivals  Shiller  question  [1979] and LeRoy and P o r t e r [1979] have a n a l y z e d the  o f whether the v a r i a n c e r a t e s i n time s e r i e s o f s t o c k p r i c e s are too h i g h to be e x p l a i n e d s o l e l y by i n f o r m a t i o n  arrivals.  For example  S h i l l e r assumes t h a t c u r r e n t s t o c k p r i c e s s h o u l d be the p r e s e n t v a l u e of optimal  f o r e c a s t s o f f u t u r e d i v i d e n d payments.  o f an o p t i m a l  f o r e c a s t i s orthogonal  S i n c e the  error  to the f o r e c a s t , - the v a r i a n c e  o f an a c t u a l d i v i d e n d s e r i e s s h o u l d be g r e a t e r than the v a r i a n c e o f the s t o c k p r i c e s .  He f i n d s t h i s bound to be too s e r i o u s l y v i o -  l a t e d to be a s c r i b e d , f o r example, to v a r i a t i o n  in interest  rates.  T h i s seems t o c o n t r a d i c t rational  the hypotheses o f market e f f i c i e n c y and  e x p e c t a t i o n s , u n l e s s one a l s o a s c r i b e s some o f the  variation  i n s t o c k p r i c e s to changes i n p r e f e r e n c e s . A n o t h e r , somewhat more c a s u a l , o b s e r v a t i o n s u g g e s t i n g t h a t p r e f e r e n c e s may v a r y o v e r time i s p r o v i d e d by van Home pp. 155-161] who notes t h a t bond r i s k premia ( i n t e r e s t  [1978, differentials  between h i g h and low grade bonds) vary o v e r the b u s i n e s s c y c l e more than i s j u s t i f i a b l e  s o l e l y by changes i n r e l a t i v e d e f a u l t  risk.  He s u g g e s t s t h a t i n v e s t o r p r e f e r e n c e s vary o v e r the b u s i n e s s c y c l e (people become more r i s k a v e r s e i n r e c e s s i o n s ) . R e l a t e d to van H o m e ' s c o n t e n t i o n i s the q u e s t i o n o f the suddenness and s e v e r i t y o f the G r e a t D e p r e s s i o n a r e a d e q u a t e l y e x p l a i n e d by the t r a d i t i o n a l  whether  really  explanations related  bank f a i l u r e s , money s u p p l y c o n t r a c t i o n s o r l i q u i d i t y  to  traps.  A sudden i n c r e a s e i n r i s k a v e r s i o n c o u l d have p r e c i p i t a t e d  the  s t o c k market c o l l a p s e . Thus, t h e r e i s m e r i t i n s t u d y i n g the q u e s t i o n o f when v a r i a t i o n s i n p r e f e r e n c e s can be d i s t i n g u i s h e d from v a r i a t i o n s given s p e c i f i c functional  in  information,  forms f o r p r e f e r e n c e s and b e l i e f s .  Another f e a t u r e of t h i s chapter i s t h a t i t  provides a construc-  tive analysis of f u l l y revealing rational  expectations  ( F R E ' s ) i n which m u l t i v a r i a t e  is revealed.  This i s  f u n c t i o n r e s u l t s o f the  previous  c o n t r a s t to the l o c a l i m p l i c i t  information  c h a p t e r s and the p u r e l y t o p o l o g i c a l  r e s u l t s of A l l e n  that e s t a b l i s h generic existence only.  equilibria in  [1978,1979],  The c h a p t e r s t a r t s w i t h a g e n e r a l a n a l y s i s f o r u t i l i t y f u n c t i o n s and a r b i t r a r y  exponential  probability distributions  p r o g r e s s e s t o more s p e c i f i c r e s u l t s f o r two p r o b a b i l i t y t h e normal and gamma f a m i l i e s . analysis in Kraus-Sick  It  and  families:  i s a generalization of  the  [1980].  The c e n t r a l q u e s t i o n o f t h i s c h a p t e r i s whether p r i c e s can reveal p r o b a b i l i t y parameters.  i n f o r m a t i o n , r i s k a v e r s i o n and i m p a t i e n c e  The aggregate r i s k a v e r s i o n parameter s t u d i e d here  i s d e r i v e d e x p l i c i t l y from i n d i v i d u a l Hdwever, o n l y a s i n g l e p r o b a b i l i t y  r i s k aversion parameters.  parameter f o r a l l  agents i s  c o n s i d e r e d , a l t h o u g h under some c i r c u m s t a n c e s , one may p r e f e r think of i t  as an aggregate s u f f i c i e n t  m a t i o n , as i n Grossman [ 1 9 7 6 ] .  s t a t i s t i c of diverse  to infor-  A l t e r n a t i v e l y , one may t h i n k o f  the v a l u e o f the parameter b e i n g exogenously r e v e a l e d to some i n formed a g e n t s , and communicated to o t h e r agents by market p r i c e s (in a FRE). S i m i l a r l y market p r i c e s depend on an aggregate i m p a t i e n c e parameter n, which w i t h e x p o n e n t i a l u t i l i t y may be thought o f as a g e o m e t r i c mean o f i n d i v i d u a l Rubinstein [1974], In the l a t t e r  o r as a commonly h e l d i m p a t i e n c e p a r a m e t e r .  c a s e , one must j u s t i f y why t h i s parameter must be  r e v e a l e d to agents i f ences.  i m p a t i e n c e p a r a m e t e r s , as i n  they a l r e a d y know t h e i r own p e r s o n a l  prefer-  The i n f e r e n c e o f n may be o f i n t e r e s t to some i n f i n i t e s i m a l  a g e n t w i t h an e n t i r e l y u n r e l a t e d p r e f e r e n c e s t r u c t u r e , who does not a f f e c t market p r i c e s but would l i k e to know more about n e x t  period  prices.  This i s the same r a t i o n a l e used i n Chapter 3, when d i s -  cussing the need f o r inference of future p r i c e s , i n the presence of the r o l l o v e r algorithm. Notation and s e t t i n g The f o l l o w i n g notation i s s i m i l a r to that used i n the two previous  chapters:  There are I agents ( i = 1 y.j c»  t  t  I)  = date t wealth of agent i ( t = 0 , 1 , 2 ) , = date t consumption of agent i ( t = 0,1,2) = agent i ' s investment i n the date t r i s k l e s s asset that matures a t date T ( t , x = 0 , 1 , 2 ; t < x )  m  it  = agent i ' s investment a t date t i n the r i s k y asset that pays o f f a t date t + l ( t = 0,1)  D  t  = date t p r i c e o f the r i s k l e s s bond, maturing a t date x ( t , x = 0 , 1 , 2 ; t < x ) .  P  t  Cj.  = date t p r i c e of the r i s k y asset ( t = 0,1) = aggregate date t dividend y i e l d e d by r i s k y asset (t = 0,1,2).  Conditional on information a v a i l a b l e a t any date p r i o r to t , any v a r i a b l e with s u b s c r i p t  t  i s , i n general, random, and t h i s  randomness w i l l be denoted by a t i l d e (~). The one period models involve dates 1 and 2 only and the two period models involve a l l three dates. Budget c o n s t r a i n t s are f o r dates 0 and 1, r e s p e c t i v e l y : < ) 5 J  ^i0  =  C  iO  + f  i01 01 D  + f  i02 02 D  +  m  i0 0 P  (5.2)  *il  = il c  +  f  il2 12 D  +  m  il l  '  P  Date 1 and date 2 wealth are r e a l i z e d as (5.3)  y„ =f  (5.4)  y  = f  i 2  +f  i Q 1  t  i l 2  D  i 0 2  .  + m (P C )  1 2  i 0  1 +  1  ^  Note that (5.3) r e f l e c t s the f a c t that the date 0 purchase of a two period bond ( f  ) yields f 2 1 2 D  i Q 2  a t  d a t e  1  a n d  t n a t  t n e  i 0  date 0 purchase of the r i s k y asset (nUg) y i e l d s m . i c  i n  c o n s u m  0  t i o n dividend at date 1 with m^P^  P  -  remaining i n c a p i t a l value.  A l l wealth i s consumed at date 2, so (5.5)  c. = y. 2  .  2  The date 0 market clears when  r  o *  0  f  ior  f  i02  (5.6)  c  0  •  m  io •  '  c  il  1  The date 1 market clears when  (5.7)  M =1  f  •  The following (Al)  r i c  ^  =  i01 m  i l  i  0 1  J  J  N  assumption i s maintained throughout:  The stochastic  process generating ( C , C-|, C ) i s a Markov Q  2  process with independent increments: A  t  s  C  t  The d i s t r i b u t i o n of  't-1  ( t = 1, 2)  i s assumed to depend- on an a r b i t r a r y parameter t,^  91.  O n e - p e r i o d models.:  Exponential  utility  T h i s s e c t i o n c o n s i d e r s markets t h a t are open a t date 1 o n l y . Assume t h a t agent i has i n t e r t e m p o r a l l y  a d d i t i v e von Neumann-  Morgenstern u t i l i t y w i t h c o n s t a n t a b s o l u t e r i s k a v e r s i o n f o r sumption a t dates 1 and 2 . (5.8)  U  where y  i 2  n  (c  i l f  ,  f  That i s  , m )  i l 2  i1;  i s a function of f..^  o p e r a t o r E„  and m ^  of C  2  by ( 5 . 4 )  of C  , -j» D^ p  2  E^expf-e.y^.  n  and the e x p e c t a t i o n  conditional  0  and C-,.  on date 1  The c o n d i t i o n a l  i s a f u n c t i o n o f the parameter r,  2  t r i b u t i o n s are c o n s i d e r e d h e r e : of  exp ( - 0 ^ ) -  ='-  i s f o r the d i s t r i b u t i o n  i n f o r m a t i o n which i n c l u d e s : distribution  con-  normal and gamma.  •  Two d i s -  The c o e f f i c i e n t  absolute^ r i s k a v e r s i o n i s 0^ > 0 and the r a t e o f i m p a t i e n c e i s  n, which i s common to a l l (A2)  n  agents.  Assume  i s known to a l l agents a t date 1.  At date 1, agent i maximizes ( 5 . 8 )  s u b j e c t to ( 5 . 2 ) and  Assuming t h a t the o r d e r o f the e x p e c t a t i o n and o p e r a t o r s can be r e v e r s e d (as w i l l o r d e r c o n d i t i o n s f o r agent i ' s  (5.4).  differentiation  be v e r i f i e d b e l o w ) , the  first  demands can be w r i t t e n as i n  Chapter 3 , as (5.9)  D  (5.10)  P  ] 2  ]  exp(-0.c ) =  n  E,;:' e x p ( - 0  exp(-0.c ) =  n  E^,C  i l  i l  2  i  y ) i2  exp(-0.y. ). 2  Note t h a t the Kuhn-Tucker-Lagrange f i r s t o r d e r c o n d i t i o n s a r e n e c e s s a r y and s u f f i c i e n t f o r an optimum because the o b j e c t i v e f u n c t i o n strictly  concave and the c o n s t r a i n t s a r e l i n e a r ( c f .  Zangwill  is [1969, c h .  2]).  At t h i s point one may use s p e c i f i c p r o b a b i l i t y d i s t r i b u t i o n s for  the expectations  and compute demand functions (at l e a s t i n  p r i n c i p l e ) and solve f o r e q u i l i b r i u m prices with the market c l e a r i n g r e l a t i o n (5.7). A l t e r n a t i v e l y , one can proceed as i n Chapter 3 and note that, i f a l l agents have the same b e l i e f s and information , the f i r s t  order  conditions and market c l e a r i n g conditions are s a t i s f i e d when  e/e,  mi l (5.11)  i l 2 c  il  =  =  f  l T D ^ i l  il2  +  m  i l  " i l m  C  ( C  1  + P  1»  l (i = 1.  (5.12)  P,  (5.13)  J  = nE  C exp(-e(C - C J )  r  2  nE  12  I)  and.  exp(-e(C - C J ) 2  ^2  where e " ^ ^ ^ ^ " ) 1  1  M u l t i p l y i n g (5.13) by C-j and subtracting from (5.12) y i e l d s (5.14)  P  1  - C  1  D  12  = E n  Agents desire to i n f e r 5  C  2  2  i n a r a t i o n a l expectations e q u i l i b r i u m .  2  2  exp(-e(C -Ci))  2  (when e i s unknown, but n> C-| are  2  known) from prices P^ and D-| R e c a l l i n g that A H  (C -d)  ? 2  - C i , i t i s apparent from (5.13) and (5.14)  that, by observing e q u i l i b r i u m p r i c e s , agents observe f(e-,c ) ~ 2  E  e K z  xp(-o  A) 2  and.  -. f e - - ( e » £ 2 ) f  E  =  A exp(-8A2) »  c  2  a s s u m i n g , a g a i n , t h a t the o r d e r o f e x p e c t a t i o n and  differentiation  can be r e v e r s e d . As e x p l a i n e d i n Chapter 2 and a p p l i e d i n Chapters 3 and 4 , the procedure f o r s e a r c h i n g f o r a f u l l y expectations equilibrium  (FRE) w i l l  informing  rational  be to assume e and £2 a r e  known to a l l , and then examine under what c o n d i t i o n s the of this  i n f o r m a t i o n can be s u s t a i n e d i n  One-period model:  equilibrium.  Normally d i s t r i b u t e d  returns  Consider, f i r s t , a normally d i s t r i b u t e d  (5.15)  A  where a  2  * N(c ,a 2  2 2  A  .  2  Suppose  )  i s assumed known.  2 2  2 2 = nexp(-ec + - - - -) e  Then and  revelation  D Pi - CiD  a  1 2  = nf(e,c )  1 2  = - - l l f ( e , * ; ) = nf(e,C2)(?2-e<?2 ) .  2  i  2  2  2  2  n  2  38  Hence, agents o b s e r v e 2 (5.16)  In  (D12/H)  =  -6c  2  2.  +  and (5.17)  P1/D12-C1  = c -ea 2  •  2 2  Given the l e f t hand s i d e s , t h e s e a r e two e q u a t i o n s i n two unknowns, 8 and c (5.18)  2  .  Substituting for  In  (n/D ) 1 2  £2 from ( 5 . 1 7 )  into  = (Pi/Di -Ci)e+(a 2  (5.16)  2 2  /2)e . 2  yields  Case a  Case b F i g u r e 5.1  S o l u t i o n of Equation  (5.18)  The r i g h t hand s i d e o f ( 5 . 1 8 )  i s a - p a r a b o l a through the  origin  t h a t opens upward, as graphed i n two p o s s i b l e cases i n F i g u r e  (5.1).  Cases  of  a  and  b  o c c u r as z e r o i s the s m a l l e r o r l a r g e r r o o t  the r i g h t hand s i d e o f ( 5 . 1 8 ) r e s p e c t i v e l y . r o o t s may o c c u r f o r most v a l u e s o f l n ( n / D advantage o f the r e s t r i c t i o n e > 0 . a g g r e g a t e i n v e s t o r be r i s k a v e r s e .  1 2  In e i t h e r c a s e , two ) , so we must take  That i s , we r e q u i r e t h a t the In e i t h e r c a s e , i t  t h a t ( 5 . 1 8 ) p r o v i d e s a unique s o l u t i o n f o r e > 0 i f f or e = ( C - P / D 1  1  1 2  )/a  2 2  and l n ( n / D ) 1 2  = - (P /D 1  is clear  either  -C ) /(2a 2  1 2  1  ln(n/D )>0 12  2 2  ).  The  l a t e r i s a s p e c i a l case where e i s a t the minimum o f the p a r a b o l a , and c l e a r l y i s n o t r o b u s t , s i n c e a minor p e r t u r b a t i o n would y i e l d two d i s t i n c t s o l u t i o n s to the problem ( a t l e a s t on must e x i s t , s i n c e the economy has an e q u i l i b r i u m ) . Since this  i s a one p e r i o d economy, uninformed agents a r e n o t  g e n e r a l l y i n t e r e s t e d i n i n f e r r i n g 0. can compute s  2  using (5.17).  However, h a v i n g computed 0,  This a n a l y s i s y i e l d s the  following:  they  Theorem 5 . 1 : all  In the o n e - p e r i o d economy, o f t h i s s e c t i o n , where  agents have c o n s t a n t a b s o l u t e r i s k a v e r s i o n and t h e same r a t e  o f i m p a t i e n c e n, and the r i s k y a s s e t has a n o r m a l l y unknown mean r e t u r n z,2  payoff with i n i t i a l l y  agents can i n f e r c,z  FRE whereby a l l  e) from p r i c e s , p r o v i d e d D  1 2  (  anc  +  distributed  Ci> t h e r e e x i s t s a  * aggregate r i s k a v e r s i o n  < n, where D  1 2  i s the date 1 p r i c e o f  one u n i t o f c e r t a i n consumption a t date 2 . It D  i s of i n t e r e s t  < n.  1 2  It  to e x p l o r e the meaning o f the  condition  says t h a t the one p e r i o d d i s c o u n t f a c t o r i s  than the r a t e o f i m p a t i e n c e .  From ( 5 . 1 6 ) , t h i s  less  is equivalent  to  2 ?2  _  6  °2  11 >  0  or  E(C ) - 6 2  a /z > 2  2  Cl-  That i s , a  FRE o c c u r s  when the c e r t a i n t y e q u i v a l e n t o f date 2 s o c i a l consumption exceeds t h e date 1 amount o f s o c i a l c o n s u m p t i o n .  (Here, the  certainty  e q u i v a l e n t i s taken w h i l e i g n o r i n g the i m p a t i e n c e p a r a m e t e r , • e f f e c t i v e l y s e t t i n g n = 1.) O n e - p e r i o d m o d e l s : Gamma d i s t r i b u t e d Suppose A  has a n o n - c e n t r a l  2  suppose the d e n s i t y o f A  2  (5.19)  returns  gamma d i s t r i b u t i o n .  i s , given a,3  P '.U > - ( V  (a>0),  (-A,+g),  2  B;  That i s ,  (  5  2  i  s )  where r ( « ) i s the i n c o m p l e t e gamma f u n c t i o n which n o r m a l i z e s the probability  to i n t e g r a t e  the parameter ? . 2  to u n i t y .  Both a and 3 a r e c a n d i d a t e s  B is a non-centrality  I c o a t i o n o f the d i s t r i b u t i o n .  for  parameter s p e c i f y i n g  G i v e n a , an i n c r e a s e i n 3 y i e l d s  an i n c r e a s e i n the mean i n c r e m e n t E (A ) 2  = E ( C ) - C i i n the s o c i a l 2  consumption p r o c e s s .  We assume  (A3)  3 > - CT_  Since A  2  .  >_ g _> - Cj_, t h i s ensures t h a t C >_ 0 .  = C 2  overcomes one o f the problems a s s o c i a t e d w i t h n o r m a l l y returns-:  This  2  they l a c k l i m i t e d l i a b i l i t y  and h o l d e r s o f t h e r i s k y  a s s e t may be f o r c e d to consume a r b i t r a r i l y consumption good - - an i n c o m p r e h e n s i b l e The mean o f A  2  distributed  n e g a t i v e amounts o f the  task.  i s a + 3 and the v a r i a n c e i s a, so a i s  s i m u l t a n e o u s l y a l o c a t i o n and s c a l i n g p a r a m e t e r . E v a l u a t i n g the moment g e n e r a t i n g f u n c t i o n . o f f  (e,5 ) = 2  A  3  yields.  exp(-e A )  E r  2  £2  = exp (-6 3 ) (1+0)" and,  2  (0>-l)  ) f (e,? ) = E 3 0 ' -  A  3  2  =  H e r e , we s e t c  2  r ?  (5.20)  1 2  ln(D /n) 1 2  Pi  exp  (-03) (l+e)"  .  -  C  P1/D12  1,1 - a;-,3  D  2  1 2  = n exp  agents observe (-03)(l+0)~  or  A  = - 03 - a £n (1+0) = (3+a/(l+0)) D  - Ci =  3  and or  1 2  + a/(l+e) .  known  S o l v i n g ( 5 . 2 1 ) f o r a and s u b s t i t u t i n g (5.22)  2  equal t o a ( o r 3 ) w i t h 3 ( o r a) f i x e d and known.  D  Case 1  exp (-0 A )  (3 + 01/(1+0))  Hence, from ( 5 . 1 3 ) and ( 5 . 1 4 ) ,  (5.21)  2  2  In (n / D  1 2  ) =  i n t o (5.20) y i e l d s  03 + (l+e)(P /D 1  1 2  - C - . B) l n ( l + e ) . x  98.  (eio)  Figure 5.3  Graphs o f F ( e ) , the r i g h t hand s i d e o f e q u a t i o n  (5.23)  99. The r i g h t side o f (5.22) i s a convex function through the o r i g i n as i n Figure 5.2. Note from (5.21) that Px/D 2-C -3>0. 1  The r i g h t  1  monotone increasing i f B>0, always  hand side o f (5.22) i s s t r i c t l y  y i e l d i n g a FRE (since (5.21).gives a i n terms of e ) .  But assuming  g>0 places strong conditions on the s o c i a l consumption process: i t requires that the s o c i a l consumption l e v e l s never decrease from one period to the next.  e  In terms o f endogenous v a r i a b l e s i t i s c l e a r that  can be i n f e r r e d (.since e>0) i f f n  as f o r Theorem 5,1.  >  Di2>  which i s the same c o n d i t i o n  Once again we note from (5.20) o r (5.13)  t h i s requires that the expected marginal  that  u t i l i t y o f date 2 consumption  be l e s s than that o f date 1 consumption disregarding impatience. marginal  u t i l i t y i s decreasing  Since  in wealth, i t means that the c e r t a i n t y  equivalent o f date 2 consumption exceeds date 1 consumption. We have e s t a b l i s h e d : Theorem 5.2:  In a one-period economy as i n t h i s s e c t i o n  (constant  absolute r i s k aversion, known impatience n, non-central  gamma d i s t r i b u -  ted s o c i a l payoff increments with known non-central ity.  parameter e),  there e x i s t s a FRE whereby a l l agents can i n f e r c 2 prices i f f D i Case 2  s  2  s 2  =  a  (  a n c  l ) f 6  r o m  n.  = 3; a known  Solving (5.21) f o r £ and s u b s t i t u t i n g i n t o (5.20) y i e l d s (5.23)  l n ( n / D i ) = e ( P i / D ^ - q )-ae/(l+0 )+a ln(l+e) 2  Let the r i g h t hand side o f (5.23) be F ( e ) . or - <*> as e-*- «=, according as P /D -C 1  zero, r e s p e c t i v e l y .  12  1  Then F(0)=0 and F(e)-> + »  i s greater than o r l e s s than  F'(e) = I V i 2 - C i ~ (i+e)*  Now,  D  W  '  so F i s asymptotically concave as e-*», and F' has the same sign a t 6=0  and e = + oo as P i / D - C . 12  1  The zeros o f F' are  I f P\/D -C >0, there are no p o s i t i v e roots of F' so F i s monotone 12  l  increasing.  If P i / D 2 " i " "  decreasing.  I f - /2<P /D -C ,  c  <  / / 2  1  a  1  12  l  ' ' has no r o o t s , so F .is'monotone F  there are two p o s i t i v e roots of F  (perhaps not d i s t i n c t ) , so F i s decreasing, then i n c r e a s i n g , then decreasing as 6 increases.  These cases are i l l u s t r a t e d i n Figure  5.3.  The f i r s t case has a FRE i f f l n ( n / D ) > 0 and the second has a FRE 12  i f f ln(n/D )<0. 12  large e.  In the t h i r d case, F i s i n v e r t i b l e only f o r s u i t a b l y  That i s , f o r some e*>0, and k<0,  F(e*)  = k, F i s monotone  on (e*,°°) and the inverse image of (-°°,k) under F i s ( e * , ° ° ) , so that F i s i n v e r t i b l e f o r e>e*, or, e q u i v a l e n t l y , ln(n/D )<k<0. 12  Theorem 5.3:  This y i e l d s :  In the one-period economy of t h i s s e c t i o n with non-central  gamma d i s t r i b u t e d s o c i a l payoff increments  there e x i s t s a FRE whereby  a l l agents can i n f e r the n o n - c e n t r a l i t y parameter s  2  = e (and e ) , from  p r i c e s , given a, i f f e i t h e r i) ii) iii)  where 0<K<1 and  n >_ D  1 2  and  P 1 / D 1 2 - C 1 >_ 0  n <  1 2  and  P1/D12-C1 1  and  - I  D  n < KD  K  i 2  i s a function Pi,  Di , 2  or, - I  or,  < Pf/Di2-Ci'<  0 ,  C i , C , n and ot. 0  Note that conditions i ) , i i ) and i i i ) are non-vacuous, f o r i ) obtains i f ot, e,  e>rj, i i ) obtains i f 0<  a  i s l a r g e , 3<_-5a (say) and  0<6 i s c l o s e to 0, and i i i ) obtains i f 3 = -ot and e>0.  The condition  n>Di2 has already been shown to be equivalent to the requirement that the c e r t a i n t y equivalent (disregarding impatience) of date 2 consumpt i o n exceeds date 1 consumption.  The c o n d i t i o n Pi-CiDi >0, f o r example, 2  i s that the value of the r i s k y asset paying o f f at date 2 should exceed the value of the c e r t a i n date 1 consumption, i f consumption i s postponed  to date 2 .  These c o n d i t i o n s sound v e r y s i m i l a r to each o t h e r ,  are mathematically q u i t e d i s t i n c t ,  but  as seen i n e q u a t i o n s ( 5 . 2 0 )  and  (5.21). Conditions i i )  and i i i )  n e c e s s a r y nor s u f f i c i e n t (5.2). it  show  that, in-general,-n>Di2 is  neither":  f o r a FRE,. which i s , the case, i n Theorems ( 5 . 1 ) and  For g e n e r a l p r o b a b i l i t y d i s t r i b u t i o n s  and p a r a m e t e r s , however,  i s r e a s o n a b l e to e x p e c t t h a t the r e l a t i o n s h i p between n and D  s h o u l d be i m p o r t a n t (5.13), D  o b t a i n e d i n the l i m i t as e-> 0 . +  the u t i l i t y f u n c t i o n  not w e l l - d e f i n e d . )  This w i l l  still  be the case i f  as a f u n c t i o n o f  e (conditional  (5.13)  to g e t an e q u a t i o n i n e a l o n e .  ways be comparing l n ( n / D ) 1 2  this  (Note t h a t  one s o l v e s  (5.12)  on p r i c e s ) and s u b s t i t u t e s This l a t t e r  equation w i l l  has a t most one s t a t i o n a r y p o i n t f o r e > 0 , a FRE w i l l  pending o n l y on the s i g n o f One p e r i o d m o d e l s :  into  al-  to a f u n c t i o n o f e t h a t v a n i s h e s a t 0 .  The f u n c t i o n may have many s h a p e s , as i n F i g u r e s 5.1 to 5 . 3 , but it  for  i s a c o n s t a n t , so p r i c e s f o r e = 0 are  for z  2  From  = n when e = 0 , a n d , under r e a s o n a b l e c o n d i t i o n s  1 2  e q u a l i t y s h o u l d be e=0,  i n d e t e r m i n i n g the e x i s t e n c e o f a FRE.  1 2  ln(n/D  1 2  if  e x i s t de-  ).  Other d i s t r i b u t i o n s  and u t i l i t y  classes  The e x p o n e n t i a l u t i l i t y c l a s s was used i n the p r e v i o u s s e c t i o n s because i t  aggregates ( y i e l d i n g parsimony o f r i s k a v e r s i o n parameters)  and because i t f(e,£ ) 2  = E  o f t e n y i e l d s c l o s e d form s o l u t i o n s to the  expectation  e x p ( - 6 A ) t h a t i s used i n e q u a t i o n s ( 5 . 1 3 ) and 2  (5.14).  For e x p o n e n t i a l u t i l i t y , one merely needs to a s s e s s the moment gener a t i n g function of A, 2  distribution  cases.  as was done i n the normal and n o n - c e n t r a l gamma  A n o t h e r approach i n v o l v e s u s i n g t h e o t h e r l i n e a r r i s k ance u t i l i t y f u n c t i o n s  toler-  t h a t p e r m i t a g g r e g a t i o n , extended power and  log:  TI  U  T'^I^II)^"Vv^J*  ( ii> ii2' ii) = c  f  m  (Of <l) Y  or  U  (Ci!»  n  f  ' ii) m  i 2  ln(e  =  + i  c  i l  )  + nE l n ( 8 . + y . ) 2  (y=0)  F o l l o w i n g the type o f development o f t h e p r e v i o u s s e c t i o n s and o f Chapter 3 , p r i c e s (5.24)  D  (5.25)  (C +0 ) 1  1 2  P  satisfy  (C +e )  1  = E  Y _ 1  A  1  (C +e )  n  = nE  Y _ 1  A  where  2  e  c 2  C (C +e ) 2  =  A  Y _ 1  A  2  Y _ 1  A  ( <1) Y  .  Here a g a i n , we have assumed t h a t t h e o r d e r o f d i f f e r e n t i a t i o n and e x p e c t a t i o n can be r e v e r s e d .  The e q u a t i o n s can be combined t o  yield (5.26)  (C +e ) 1  Y _ 1  A  (P +e D ) 1  A  = E  1 2  n  ? 2  (C +e ) 2  A  Y  .  In t h e c a s e o f l o g u t i l i t y , y 0 and ( 5 . 2 6 ) becomes =  p  i + e D i 2 = n(C +e ) A  1  which can be s o l v e d f o r e i n t o (5.24)  A  A  p r o v i d e d n|=Di . 2  Substituting y = 0  yields  h  2  ( C  2  + e  A  )  =  n(C  1 +  e ) A  The r i g h t s i d e i s known a t date T and t h e q u e s t i o n i s whether ?  2  can be i n f e r r e d from t h i s e q u a t i o n . d e c r e a s i n g convex f u n c t i o n o f C the d i s t r i b u t i o n s  Note t h a t ( C + e ) 2  so t h a t i f  2  i n accordance w i t h s t r i c t  1  A  is a  the parameter z,2  second degree s t o c h a s t i c  dominance ( s e e , f o r example, Hanoch-Levy [1969]) then i t i n f e r r e d from p r i c e s .  T h u s , f o r example, i f  parameter such as the mean, i t FRE e x i s t s w i t h l o g  ranks  ?  2  i  s  a  can be  location  can be i n f e r r e d from p r i c e s and a  utility.  To s t a t e these r e s u l t s more f o r m a l l y we must e x p l i c i t l y the assumption o f the i n t e r c h a n g e a b i l i t y  state  o f e x p e c t a t i o n and d i f -  ferentiation: (A4)  Interchange o f d i f f e r e n t i a t i o n  Assume f o r a l l  ^  t h a t  and  2  that  C  2  (mC  W h  and  o r  £  A  E  g|-'E  C 2  { 2  2  ^  +  f  f)  )  - E  Y  y  =  ? 2  h  2 +  2 +  f)  Y  (Ofr<l)  y  E .3^1n(mC +f) 5 2  ln(mC +f) = E 2  ^(mC  expectation.  W^^2+f)  2  ln(mC f) =  and'  2  g|^.ln(rnC +f)  (y=0)  2  V e r i f i c a t i o n o f (A4) can be done, a t l e a s t i n  principle,  a f t e r making s p e c i f i c assumptions about the d i s t r i b u t i o n and then e i t h e r e x p l i c i t l y  differentiating  dominated convergence theorem.  .  (family)  o r u s i n g the Lebesque  C l o s e d form s o l u t i o n s f o r  l o g u t i l i t y and i t s d e r i v a t i v e s f o r i n t e r e s t i n g  expected  probability  families  are r a r e ,  so i t  is difficult  e s t a b l i s h the v a l i d i t y Theorem 5.4  o f (A4) o r i l l u s t r a t e  2  the following  C o n s i d e r a one p e r i o d economy as i n t h i s  where t h e d i s t r i b u t i o n by c .  t o p r o v i d e n u m e r i c a l examples t o  Suppose t h a t ?  o f date 2 s o c i a l w e a l t h C 2  < ?2* i f f C  2  9  e n e r a  '  dominates (second degree) C g e n e r a t e d by ? . have extended l o g u t i l i t y , reveal s  2  and e^,  if D  Two p e r i o d m o d e l s :  1 2  section  i s parameterized  ' by  t e c  2  2  2  theorem.  stochastically  Then i f a l l agents  a FRE e x i s t s whereby date 1 p r i c e s  * r\.  Exponential  utility  The s o l u t i o n s t o t h e g e n e r a l model a t date 1 , as e x p r e s s e d i n equations ( 5 . 9 ) to (5.14)  can be used t o o b t a i n a d e r i v e d  u t i l i t y f o r date 1 w e a l t h y ^ , which c a n then be used a t date 0 to o b t a i n date 0 p r i c e s .  S o l v i n g (5.11)  f o r c ^ and n o t i n g ( 5 . 9 )  y i e l d s t h e f o l l o w i n g e x p r e s s i o n f o r date 1 d e r i v e d u t i l i t y f o r ' agent i (5.27)  (cf.  (5.8)): U  il^il>  -  = -  (l+D )exp(-e. 1 2  C i l  )  :y :-(e/e )(Ci+Pi) il  (1+Di )exp  i  2  1+D 12 + (0/6.)C  1  6• (1+D )<j) e x p ( - 7 + 0 ^ ^ 1 1 ) 12  f o(Pi-Di2 i) ) C  (5.28)  exp  1+D  12  w  h  e  r  e  In g e n e r a l , as viewed a t d a t e 0 , <>j i s random, s i n c e P i , D and  Ci a r e random.  However, u s i n g assumption ( A l ) , which i s  the i n c r e m e n t s A ^ =  t  shows t h a t P i - D ^ C x  n and e a r e known) s i n c e i t  t r i b u t i o n of A  2  that  - C _-j o f the s o c i a l consumption p r o c e s s  are independent, equation (5.14) a t date 0 ( i f  i 2  is non-stochastic  depends o n l y on the d i s -  about which agents have the same " i n f o r m a t i o n  date 0 as a t date 1.  S i m i l a r l y , i n (5.13)  o n l y on the d i s t r i b u t i o n  of A  2  if  the value of D  1 2  at depends  n and e a r e known, so t h a t i t  a l s o n o n - s t o c h a s t i c a t date 0 , s i n c e we a r e o n l y i n t e r e s t e d i n i n w h i c h date 0 p r i c e s r e v e a l n and 6 (and S i o r 5 2 ) .  is FRE's  Note t h a t  t h e c o n s t a n t a b s o l u t e r i s k a v e r s i o n ( w i t h no w e a l t h e f f e c t s on p r i c e s ) and  t h e assumption o f a Markov s o c i a l consumption p r o c e s s w i t h  independent i n c r e m e n t s combine to m a k e ; D i ( a n d f u t u r e s p o t  interest  2 :  rates) deterministic.  Adding i n e x p o n e n t i a l  utility  f o r date 0  consumption and d i s c o u n t i n g the d a t e 1 d e r i v e d u t i l i t y by the f a c t o r n y i e l d s date 0 u t i l i t y (5.29)  of  ib1 i0' i01 ' g  f  f  i02' i0^  = -  m  exp(-0c  i o  + n E  )-TiE  ? 2  ? i  (exp(-e c i  exp(i-e.c. )) 2  = - exp(-e.c )-n<(>(l+Di ) 2  i0  e 1+D  where y - ,  satisfies  (5.3).  1 1  )  Since d  1 S l2  d e t e r m i n i s t i c a t d a t e 0 , t h e pure e x p e c t a t i o n s  v e r s i o n o f t h e term s t r u c t u r e o f i n t e r e s t r a t e s must h o l d ( i n t h e absence o f a r b i t r a g e )  so t h a t  (5.30)  02  D  01 12  = D  '  D  T h i s a l l o w s agents t o s e t f . Q = 0 w i t h o u t l o s s o f g e n e r a l i t y , 2  s i n c e t w o - p e r i o d r i s k l e s s i n v e s t m e n t s can be o b t a i n e d by r o l l i n g o v e r a o n e - p e r i o d bond a t d a t e 1. Assume CQ i s exogenously  r e v e a l e d t o a l l agents a t d a t e 0 .  Maximizing (5.29) aubject t o f j demands f o r C ^ Q , yields prices.  0 2  = 0 and ( 5 . 1 ) and ( 5 . 2 ) y i e l d s  f IQ-J , and m ^ - ] . U s i n g ( 5 . 6 ) t o c l e a r t h e market In e q u i l i b r i u m we h a v e :  m m  i0  f  i  0  c  i0  p  0  _ e. _ e / ( l + D : ) eT " eTAtTFTJ^) 12  =  =  =  (D  0 1  /(l+D  W * ^" 1  0  1  +  m  C  1  ? I  1  1  1 2  1  (1+D12)C0))].  = n<(>E [exp(-6(l+D )" (C +P Ci  12  -  1  (ltD12)Cb))]  Now, (5.31)  0»  i0 0  1  Q 1  + P  = n<l»E [(C +P )exp(-e(l+D )" (C +P -  D  rVWV 0 C  1 2  CT+PT-O+D^CQ = (P1-D-|2C1) + ( l + D ^ ) ^  1  1  and we have seen t h a t is D  1 2  -  p  -|  - D  i2  C  U s i n g the d e f i n i t i o n  s i m p l i f y to  l  l  s  n o n - s  t o c n a s  ' '' t  c  a  t  d  a  t  °»  e  a  s  (5.28) o f ty, t h e p r i c e e q u a t i o n s  '  (5.32)  P  (5.33)  D  =  0  nE [(Ci+Pi)exp(-9A )] Ci  ( i . e . , Ci=s )  1  = nE [exp(-6A!)]  Q 1  .  ?i  Note from (5.13) and ( 5 . 3 3 ) distributed  '  i  t h a t , i f Ai and A  are i d e n t i c a l l y  2  a d d i t i o n to b e i n g i n d e p e n d e n t , D = D  n  Q 1  2  I f t h i s were t h e case then t h e date 0 p r i c e s P , D 0  and D  0 1  Q 2  1 2  would  2 o n l y p r o v i d e two p i e c e s o f i n f o r m a t i o n s i n c e D = D 0 2  which i s n o t enough t o f u l l y  0 1  r e v e a l a l l o f e » n a n d c,l  (by 5 . 3 0 ) , o r z,2 •  Hence we must assume t h a t £ 1 ^ 2 and i n f e r one o r t h e o t h e r p r o b a bility  parameter o n l y , i n o r d e r t o o b t a i n  FRE's.  I t i s c o n v e n i e n t to use ( 5 . 3 1 ) and ( 5 . 1 4 ) P  =nE  Q  ? i  to r e - e x p r e s s  [ ( - n ^ - f (e , c ) + ( l + D 2  12  (5.32)  )(C +Ai)) 0  exp(-6A!)] where  f( ,?2) = E exp(-6A )  L e t t i n g g (0 , S i ) = E  e  ?2  ? i  2  .  expt-eAj.), t h i s becomes, a f t e r  interchanging  the o r d e r o f e x p e c t a t i o n and d i f f e r e n t i a t i o n ,  (5.34)  P = (-n Q  2  ^  f(e,c ) n(l+D )C ) 2  +  " n O + D ) z~ 12  1 2  0  9(6,^)  g(e, ) ? 1  108.  Two period models:  Normally distributed returns  Suppose that  There are two candidates for the probability parameter that is to be revealed: will  z, [t, known) and ^ 2  (s  1  be analyzed here.  known).  2  The case where £ i = c  (a stationary process for A^.) w i l l  Both cases  and ° t h are unknown D  2  not y i e l d a FRE since we have  seen that i t forces D = D , eliminating one dimension from the 0 1  1 2  price information. We have g-.(e,ci) = E  ? i  exp(-6A ) 1  2  = expC-e^+e oi /2.)  and  2  f ( e , c ) = exp(-e? e cr /,2) so, from (5.33), (5.34) and (5.16), +  2  2  2  D i = ng(e ?!) (5.35) (5.36)  :ln(D /n) 01  P  = -e?i +d a /z 2  and  1  l  = C - n ( - ? 2 e o 2 ) ( e ' S 2 ) n ( l D i 2 ) C ] g(e>Ci) 2  0  or  9  0  ,  2  2  +  2  f  +  +  0  - n(l+D )(-?i+ea )g(e,?i) 2  12  1  = DoiLD^^+d+D^J^^Di^^d+DizJa^ie  + (1+D )C ] . 12  0  At date 0, agents observe P , D Q  and D  q 1  or e q u i v a l e n t l y ,  Q2>  from (5.30), they observe P , Dg-, and D . Q  These p r i c e s are  12  given by (5.36), (5.35) and (5.16).  E l i m i n a t i n g n from (5.35)  and (5.16) y i e l d s ln(Di /D ) = e ( ^ - ) + 6 (a -a )/2 .  (5.37)  2  2  0 1  2  ? 2  2  2  1  We assume a l l agents i n i t i a l l y know Co and c ^ and o 2  2 2  .  Case 1 ? known; d , n, e unknown. 2  Solve (5.36) f o r z i n terms o f e and s u b s t i t u t e into (5.37) 1  to get (5.38)  ln(D /D i) = eCO+D^r^Po/Doi-D^-O+DiaKo)-^] 1 2  0  + 9 [(a -a )/2. 2  2  2  1  2  + (l+D ) (D a _ 1  1 2  1 2  + (1+D )a ))]  2  2  2  1 2  1  =e[(l+D )" (P /D -D c: -(l+D )c )-c: ) 1  1 2  +  6  2  [ ( 3 /  0  2  )  2 C  1 2  0 1  l  2  1 2  0  2  +;i/2(D -l)/(D +l)a 12  The r i g h t hand side o f (5.38) i s a parabola that-.opens  12  ]  2  up or down,  according as 3a +(D -l)/(D +l) a 2  1  12  i s p o s i t i v e or negative, r e s p e c t i v e l y .  12  2  As i n the a n a l y s i s o f  Theorem 5.1 i n Figure 5.1, t h i s y i e l d s a FRE according as l n ( D i / D i ) 2  i s p o s i t i v e or negative, r e s p e c t i v e l y . Theorem 5.5  0  This y i e l d s :  Consider a two period economy where aggregate con-  sumption C^ follows a Markov process with independent,  normally  d i s t r i b u t e d increments, as i n t h i s s e c t i o n . Agents have exponential  u t i l i t y w i t h aggregate r i s k a v e r s i o n 0 and i m p a t i e n c e parameter If  t h e second p e r i o d mean increment E(A )  = e  2  r ]  .  known, b u t the  1 S 2  f i r s t p e r i o d i n c r e m e n t t,\ i s unknown, then a F R E e x i s t s whereby date 0 p r i c e s o f t h e r i s k y a s s e t and the bond term s t r u c t u r e v a l u e s o f e,n and s  l  s  f o r e>0, i f f  reveal the  either 2  i)  and  D 1 2 > Doi  3a + ( D 2  x  1 2  -l)/(D  1 2  l)a > 0  +  2  or  2  ii)  and  D12 < oi D  3a! + ( D 2  1 2  -l)/(D  l ) a < 0.  +  1 2  2  To  check t h a t c o n d i t i o n s i ) and i i ) a r e not v a c u o u s , note  i)  occurs i f  occurs i f  c  2  Zi>^z>0 and a  2 1  i s l a r g e and a  =a 2  2 2  that  i s a p p r o p r i a t e l y l a r g e , and i i )  and n a r e s m a l l so t h a t . D i ' « T and £ i < c , 2  2  2  a i > a , f o r a given 0 > 0. 2  2  2  Case 2  z\ known; ^ > 2  Solve (5.36) (5.39)  n> e unknown.  f o r <; and s u b s t i t u t e  (5.37)  to get  ln(D /D ) = 0(c -Po/(D iDi )-(l+D " )(c +Co)) 1  1 2  1  0 1  -  The  into  2  O  6 (a  2  2  2  2  1  1 2  /2+( / +D 3  2  " )a 1  1 2  2 1  )  r i g h t hand, s i d e i s a downward opening p a r a b o l a from which e > 0 c a n be :  inferred i f f Theorem 5.6  D <D . 1 2  0 1  This y i e l d s :  In a two p e r i o d economy as i n Theorem 5 . 5 , e x c e p t t h a t  the f i r s t p e r i o d increment Z\ i s known, and s , r\ and 0 a r e unknown, 2  a F R E e x i s t s t h a t r e v e a l s t h e s e parameters i f f It  is of interest  1 2  <D  0 1  to s t u d y the c o n d i t i o n D < D Q I .  t h a t the one p e r i o d f o r w a r d r a t e o f i n t e r e s t rate of i n t e r e s t ,  D 1 2  .  T h i s says  s h o u l d exceed t h e s p o t  o r , e q u i v a l e n t l y , t h a t t h e term s t r u c t u r e  i n t e r e s t r a t e s s h o u l d be r i s i n g .  of  Ill,  Two period models:  Gamma distributed  returns  In this s e c t i o n , the increments to social wealth, A  and A i  2  both follow the non-central gamma d i s t r i b u t i o n given in (5.19). Howevery.we shall suppose that the parameters a and 3 vary from date 1 to date 2.  That i s , assume the densities of A]_ and A  (A -B ) 1  P«l,&l  1  (Ai>Bi)  2  2  Hence  1  (Ai) = M  (A -3 )  Pa ,3  (A )  exp(-A +3 ) ~  2  2  2  =  2  2  =  2  (A >B ) 2  (a )  r  f(e»s )  are  exp(-A +6 )  a i _ 1  1  r  and  2  2  2  . exp(-6A )  E  2  2  g(e,ci)  =  exp(-eB )(l+e)"  =  E  and  a2  2  =  ?  exp(-GAi)  exp(-6B )(l+e)"  ai  1  so that, from (5.33), (5.34) and (5.20), DQI =  (5.40)  ln(D /n)  =  01  (5.41)  P  0  =  n g(e Ci) -93 -a ln(l+e)  +  f  and  1  1  [n(32W'( e)) (Oi?2) ( 1 +  or  2  ; 1 + D  i2) o3ng(9 ci)' c  1  + (l+D )[3 + /(l+e)]g(e, ) n  =  1 2  1  a i  C l  D [D 2(32 a /(l+.e)) + (l+Di )C +(l+D )(g +a /(l+e))] +  0 1  1  2  2  0  1 2  1  1  Equation (5.20) becomes, in this (5.42)  notation,  ln(D /n) = ln(E 12  exp(-8A)) 2  r  = -eg -a ln(l+e) 2  2  Subtracting .(5.'40) from (5.42) y i e l d s (5.43)  l n ( D / D ) = e(g -3 )+ln(l+e)(a -a ) 1 2  c l  1  2  1  2  .  These are four possible candidates f o r the unknown probability parameter:  a  l 9  a , p 2  and g .  x  2  T h i s . y i e l d s four cases to  analyze where one parameter i s unknown (along with e and n) and the other three are known (along with C ) .  The cases w i l l be analyzed  0  and summarized in one theorem. Case 1  t,\  = ai unknown; a , g 2  Solving (5.41) f o r  l s  g , known. 2  a and substituting into (5.43) y i e l d s  ln(D /D ) = e(3 -B ) 1 2  0 1  1  2  By (5.41), the factor in square brackets i s , in equilibrium  Di a (l Di2)oi 2  +  +  > 0,.-  2  T+e  so that the right hand side is a~convex function of e which vanishes at the o r i g i n and i s increasing f o r large  e.  However,  i f 3i<6 , the function may be decreasing f o r small e. The 2  analysis i s the same as for Case 1 in the one-period gamma distributed returns model (Theorem 5.2) as i l l u s t r a t e d in Figure 5.2,  so that a FRE exists i f f l n ( D / D i ) > 0 . 12  0  113.  Figure 5.4  Graphs of F ( e ) , the r i g h t hand side of (5.44)  Case 2  c  =  a  2  2 unknown;  S o l v i n g (5.41) (5.44)  l  3i, B  s  known  2  for a /and substituting  ln(D  A g a i n , the  ,a  i n t o (5.43)  2  1 2  /D i)  = 9(B -g .)  0  1  2  term i n square b r a c k e t s i s p o s i t i v e , and the  hand s i d e i s i n c r e a s i n g f o r l a r g e e (and f o r a l l although i t if  yields  i s not n e c e s s a r i l y convex f o r a l l  (2+Di ~ )a  i s l a r g e and the f a c t o r  1  2  1  s m a l l , the r i g h t s i d e of r i g h t side of illustrated  (5.44)  (5.44)  (For example,  9>0.  i n square b r a c k e t s  i s hot c o n v e x . )  If  Denoting  is the  graph.: F i ( e ) o r F s ( 9 ) o c c u r s ,  i s n e c e s s a r y and s u f f i c i e n t  2  3i>B2)>  by F ( 0 ) , t h r e e p o s s i b l e graphs o f F ( 9 ) a r e  in Figure (5.4).  ln(Di /Doi)>0  e if  right  o c c u r s , then t h e r e e x i s t s a k> 0 and 0*>O  f o r a FRE.  I f F2(9)  such t h a t F ( e * ) = k,  F i s monotone on [9*,°°) and the i n v e r s e image o f [ k , ) under F 00  is [e*,~). that  k  That i s , a FRE e x i s t s i f f  ln(D /D i)>k>0. 1 2  Note  0  depends, i n g e n e r a l on the known o r observed parameters  Po> D Q I , D]_ , C O , 04, g . B . 2  Case 3  l 9  2  C i = B i unknown, a i , a , g 2  2  known.  U s i n g the t e c h n i q u e s o f Case 1 and Case 2 y i e l d s the  (5.45)  ln(Di /Doi) = 2  0  -(l+D )Co-(l+2Di )B ] 1 2  1+Di  2  2  2  equation  115.  F i g u r e 5.5  Graphs o f ; G ( e ) , the r i g h t hand s i d e o f for  F i g u r e 5.6  ai>a  2  Graphs o f G ( e ) , the r i g h t hand s i d e o f for  ai<a  2  (5.45)  (5.45)  L e t t i n g Y be t h e f a c t o r i n square b r a c k e t s , i n e q u i l i b r i u m , by (5.41),  i t must be t h a t Y n + D i 2 ) ( B i - B ) + ( D i 2 a 2 + ( l + D i 2 ) o n ) / ( l + e ) . =  2  T h i s may be p o s t i v e o r n e g a t i v e , b u t tends t o be p o s i t i v e as « i , a > 0 . 2  Thus, i f t h e r i g h t s i d e o f ( 5 . 4 5 )  may be t h a t G(e)  ±°° as e  insofar  i s G(e), i t  » . A l s o , f o r 0>O, t h e second term o f  G(e) i s convex d e c r e a s i n g , and t h e t h i r d term i s concave i n c r e a s i n g o r convex d e c r e a s i n g , a c c o r d i n g as ai>a  2  S i n c e t h e second term  or a < a , respectively. 1  2  i s bounded, G(0) i s convex o r concave  f o r l a r g e 0 a c c o r d i n g as t h e t h i r d term i s convex o r c o n c a v e , a l t h o u g h G may have any c u r v a t u r e f o r s m a l l p o s i t i v e 0 i f t h e second and t h i r d terms have d i f f e r e n t illustrate  curvature.  p o s s i b l e graphs o f G ( 0 ) .  i s a s y m p t o t i c a l l y concave as 6 as Y i s p o s i t i v e o r n e g a t i v e .  F i g u r e s 5 . 5 and 5 . 6  I n F i g u r e 5 . 5 where a > a , G 1  and i n c r e a s i n g o r d e c r e a s i n g T h u s , i f ai>c<2. a FRE e x i s t s  Y>0 and l n ( D i 2 / D o i ) > 0 o r i f Y<0 and l n ( D i / D r j i ) < k / < 2  function of P  l  s  2  Doi, D i , Co, <*i, a 2  2  and B . 2  where a ! < a , G i s g l o b a l l y concave ( f o r 0>O). 2  if  0 where k i s a  In F i g u r e 5 . 5 In t h i s c a s e , a  FRE e x i s t s i f f Y>0 and l n ( D i / D i ) > 0 o r y<0 and l n ( D i / D i ) < 0 . 2  Case 4  S  2  = $  2  0  unknown; 04 , a  E l i m i n a t i n g &z from (5.41)  ln(Di2/Doi) =  Dirti^T  2 i  2  ,3  2  0  known.  and ( 5 . 4 3 )  yields  -( .Pi2)Co-3i]- ^-{a2+(l+Di " >i} 1+  1  1  2  •+ln(l+0)(a -a ) 1  2  C o n d i t i o n s f o r FRE  Unknown P r o b a b i l i t y Parameter  D12 > D i 0  °i  D12  0-2  K2D01  >  a >a2» 1  Y>0 and D  1 2  >D i 0  ai>a > y<0 and D < K 2  1 2  3  a < a . Y>0 and D > D Q I 1  2  1 2  a < a . Y<0 1  3  2  n  d  D  Doi  < 1 2  a!>a » K>0 and D 2  2  1 2  >D i 0  a!>a )  K<0 and D < K D  a <a2»  K>0 and D  1 2  >D  a2<ci2> K<0 and D  1 2  <D i  2  1  Note:  a  12  )+  0 1  0  K >1; K  2  depends upon 0 4 , 3  0<K <1; K  3  depends upon  ot]_>  C42>  0<K <1; K  4  depends upon  ot 19  a > 3 i > p > Doi» D > C  2  3  1+  Y = P0/D01 -  (1+D )C  0  < = PQ/DOI -  (1+D )C  0  Table 5.1  12  1 2  -  l  9  f3 »  Doi» D » C  32»  D  12  2  2  0  (i+2D12)e2  C o n d i t i o n s f o r a FRE i n Theorem 5.7  1 2  1 2  J C  0  0  0  118  L e t t i n g K be t h e f a c t o r i n square b r a c k e t s , i n e q u i l i b r i u m , K = D (B +B ) +(D o + (l+D )a )/(l4e), 1 2  2  1  1 2  2  1 2  1  and t h i s may be p o s i t i v e o r n e g a t i v e .  The a n a l y s i s i s e x a c t l y the  same as f o r Case 3 , w i t h K r e p l a c i n g y. These cases may be summarized i n t h e f o l l o w i n g Theorem 5 . 7 :  theorem.  C o n s i d e r the two p e r i o d e x p o n e n t i a l u t i l i t y economy  where the s o c i a l w e a l t h s t o c h a s t i c p r o c e s s has independent n o n - c e n t r a l gamma d i s t r i b u t e d -increments as i n t h i s s e c t i o n . observe p r i c e s P , D 0  0 1  , D  (and hence D  0 2  1 2  A t d a t e 0 , agents  ) , and attempt t o i n f e r e ,  n and one o f a , a , 3 , e 2 , which a r e p r o b a b i l i t y x  2  :  FRE's o c c u r under the c o n d i t i o n s i l l u s t r a t e d Two p e r i o d m o d e l s :  Portfolio  parameters.  Then  i n Table 5 . 1 .  rollovers  In a l l o f these m o d e l s , agents have the same h o l d i n g s o f the r i s k y a s s e t a t d a t e 1 as a t date 0 .  That i s , m^Q = e/e^ = m ^ .  s u g g e s t s t h a t some s o r t o f p o r t f o l i o  r o l l o v e r t e c h n i q u e might make i t u n -  n e c e s s a r y to i n f e r t h e t a s t e and p r o b a b i l i t y In t h i s c a s e , however, agents w i l l the p r o b a b i l i t y  wealth.  p a r a m e t e r s , as i n "Chapter 3 .  always d e s i r e to know something about  parameter, s i n c e i t a f f e c t s  (and perhaps i n t e r m e d i a t e )  their u t i l i t i e s  and a l l r e c e i v e d t h e same updated i n f o r m a t i o n  t h a t a l l agents w i l l  In t h e s e m o d e l s ,  parameter i s a v a i l a b l e a t date 0 , so  p r e f e r to use i t as soon as i t i s a v a i l a b l e .  some agents w a i t u n t i l will  parameters  a t date 1 , so t h a t t h e  from d a t e 0 t o date 1.  i n f o r m a t i o n about the p r o b a b i l i t y  for final  In t h e models o f Chapter 3 , a l l  agents were e q u a l l y informed a t date 0 about the p r o b a b i l i t y  c o n t r a c t curve d i d not s h i f t  This  date 1 to use the i n f o r m a t i o n ,  If  the c o n t r a c t c u r v e  s h i f t a d v e r s e l y f o r - t h o s e a g e n t s , .and any r o l l o v e r a l g o r i t h m must  fail  Conclusion In t h i s c h a p t e r , one and two p e r i o d exchange economy models were s t u d i e d where agents had i n t e r t e m p o r a l l y  additive  u t i l i t y , d i s c o u n t e d by an i m p a t i e n c e parameter n • on n, an aggregate r i s k parameter.  aversion  parameter e  exponential  P r i c e s depend  and a  probability  C o n d i t i o n s a r e d e r i v e d , t h a t a r e v e r i f i a b l e because  they i n v o l v e o b s e r v a b l e p r i c e s and known p a r a m e t e r s , which ensure the e x i s t e n c e o f a f u l l y i n f o r m i n g (FRE).  In a F R E , agents can i n f e r  rational  expectations  6 and a p r o b a b i l i t y  equilibrium parameter  from the p r i c e o f a bond and a r i s k y a s s e t i n a o n e - p e r i o d m o d e l , o r n,; e and a p r o b a b i l i t y  parameter ; from the p r i c e s i n a term  s t r u c t u r e o f bonds and the p r i c e o f a r i s k y a s s e t . C o n d i t i o n s f o r a FRE i n c l u d e , i n t e r a l i a , the  relationship  between n and the p r i c e o f a d i s c o u n t bond, i n the o n e - p e r i o d models and the s l o p e o f the term s t r u c t u r e o f bond y i e l d s , i n the period models. central  The models used n o r m a l l y d i s t r i b u t e d  gamma d i s t r i b u t e d  two-  and non-  increments to s o c i a l w e a l t h .  Another o n e - p e r i o d m o d e l , was found to have a FRE, i n which a g g r e g a t e r i s k t o l e r a n c e and a p r o b a b i l i t y inferred  from p r i c e s , i f  family of d i s t r i b u t i o n s  the p r o b a b i l i t y  utility.  parameter ranked the  i n the same o r d e r as would s t r i c t second  degree s t o c h a s t i c dominance. log  parameter c 2 c o u l d be  Agents i n t h a t model have extended  Footnotes to Chapter 5  1.  E q u i l i b r i u m prices are unique because they can be derived by p o r t f o l i o separation from a complete market s t r u c t u r e (with a continuum o f markets) i n which prices are unique because o f the aggregation  properties o f exponential  u t i l i t y . (Cf., Footnote 2,  Chapter 3 ) . 2.  To v e r i f y the interchange o f d i f f e r e n t i a t i o n and expectation, one could e i t h e r t r y to f i n d a measurable bound f o r the d i f f e r ence quotients and use the Lebesgue dominated convergence theorem (cf.,  Rudin [1964, p. 246]) or a c t u a l l y perform the i n t e g r a t i o n  and d i f f e r e n t i a t i o n .  While the former technique i s general and  may work well f o r a v a r i e t y of u t i l i t y f a m i l i e s , i t i s not c l e a r that i t works with exponential  utility.  If A ^ N(y,c ) then 2  and  E(exp( A))  exp( a /2+y )  j Eexp( A)  ( a +y)exp( V/2+y )  2  Y  d  Y  2  Y  Y  2  Y  Y  Y  Y  On the other hand, E(-r-  exp( A)) =  EAexp( A)  Y  Y  A e x p ( A - ( A - y ) / ( 2 a ) ) dA 2  2  Y  s  exp( y+a Y  2  2 Y  = [exp( y+a Y  /2) f~ exp(-( -(y+a )) /(2c )) 2  A  2  2 Y  A  /2)](y+a ) , 2  Y  2  Y  2  dA  121.  since the l a s t i n t e r g r a t i o n i s the same as that performed to c a l c u l a t e the mean o f a normal v a r i a t e . Thus, the order of d i f f e r e n t i a t i o n (w.r.t. c o e f f i c i e n t s o f ) and expectation A  2  can be reversed.  With expoential u t i l i t y , d i f f e r e n t i a t i o n  with respect to f  i n exp(- (mA+f)) presents no problem since e  the f a c t o r exp(-of) may be taken outside the expectation. To v e r i f y t h e - v a l i d i t y of the interchange o f d i f f e r e n t i a t i o n and expectation, suppose A i s a c e n t r a l gamma v a r i a t e with density (A>O) .  A "exp(-A)/r(a) a  The r e s u l t s when ^ i s a non-central gamma v a r i a t e follow.by t r a n s l a t i n g a c e n t r a l v a r i a t e . Then ./ 7\ —9> Eexp(-yA) = —3(/ -1,+ Y )\ dy  f, \- -I = -a(l+y) a  a  9Y  (1+Y>0) . On the other hand,  E-^-exp(-YA) = E ( - A x p ( - Y A ) ) e  -3y  = -(r(a))-  ,1  '• a- 1 Aexp(-yA)A exp(-A)dA 0  = -(r(a)- ( +l)- 1  a  Y  1  (( +l)A) exp(-(Y+l)A)d[(Y+l)A] a  Y  ^ 0  =  -(Y+l)" " r(a+l)/r(a) a  = -a( +l)"  ]  a _ 1  Y  This j u s t i f i e s ^ t h e interchange o f expectation and d i f f e r e n t i a t i o n .  122.  4.  It  i s i n t e r e s t i n g to check (5.32) and (5.33) by a n a l y z i n g the  m a r g i n a l r a t e s o f s u b s t i t u t i o n f o r the aggregate The  investor.  m a r g i n a l r a t e o f s u b s t i t u t i o n between a p r o p o r t i o n a l  claim'  t o d a t e 1 and 2 consumption and i n c r e m e n t a l consumption a t date 0 f o r the aggregate i n v e s t o r E  QTTE  r  is:  [C e x p ( - 6 C )+nC exp(-GC  )]  eexp(-eCo)  = nE  = nE  [Ciexp(-e(Ci-C ))+nE 0  r  C exp(-e(C -C ))] 2  2  0  Ciexp(-eAi)+ 2'E -exp.(-e(C -.Co.))E C exp(-e(C -Ci)) i Si. - s n  r  r  1  z  2  2  2  = nE  ~  Giexp(-0A )+n E  exp(-6Ai)E  1  ?i  Si  = nE^ Ciexp(-eA )+nP E , exp(-6Ai) i  1  1  j  i  T h i s i s the same as the e x p r e s s i o n (5.32) The  .:c exp(-e(a )) 2  Q  2  2  using (5.12)  for P . 0  m a r g i n a l r a t e o f s u b s t i t u t i o n between c e r t a i n  consumption a t d a t e 1 and ( i n c r e m e n t a l )  (incremental)  consumption a t date 0  is eE n  exp(-eCi)  —  = nE  =  eexp(-eC ) 0  which i s the same,as (5.33)  for  D  0 1  .  .  exp(-QA ) l  5 1  ,  Chapter 6  Concluding  Remarks  Several questions have been asked, and at l e a s t p a r t i a l l y answered i n t h i s t h e s i s .  They include:  What i s a r a t i o n a l expectations equilibrium? Drawing upon d e f i n i t i o n s already i n the economics l i t e r a t u r e , information i s defined, i n Chapter 2, i n terms of a-algebras p a r t i t i o n s of the states of nature, upon which agents may conditional expectations.  or  form  Information i s d i s t i n g u i s h e d  from b e l i e f s , the l a t t e r being an agents' view of the p r o b a b i l i t y d i s t r i b u t i o n of a l l possible states of nature.  This i s perhaps the  f i r s t paper to s t r e s s the d i f f e r e n c e between information and b e l i e f s i n the theory of r a t i o n a l expectations.  Macro-economic  models tend to stress the importance of agents forming "correct" b e l i e f s i n r a t i o n a l expectations models.  Micro-economic models,  l i k e those i n t h i s paper, tend to concentrate on the information aspects i n the theory of r a t i o n a l expectations, while merely making convenient assumptions about b e l i e f s . A r a t i o n a l expectations e q u i l i b r i u m i s characterized as a f i x e d point i n a function space of p r i c e random v a r i a b l e s .  Existence  of e q u i l i b r i u m i s affected by at l e a s t two sorts of problems: d i s c o n t i n u i t i e s i n demands induced by d i s c o n t i n u i t i e s i n information (the c o n t i n u i t y problem), and the requirement that agents' excess demands, which may be measurable with respect to d i f f e r e n t a-algebras, must sum to a non-random constant i n e q u i l i b r i u m (the measurability problem).  H e u r i s t i c arguments advanced here, .  c o n c e r n i n g the m e a s u r a b i l i t y p r o b l e m , s u g g e s t t h a t the e x i s t e n c e of rational (ie.,  e x p e c t a t i o n s e q u i l i b r i a t h a t a r e not f u l l y  informing  i n which a l l agents do not have the same i n f o r m a t i o n  they o b s e r v e p r i c e s ) may be a r a r e o c c u r r e n c e .  An i n t e r e s t i n g  f o r f u t u r e r e s e a r c h would be to study the r o b u s t n e s s o f existence of equilibrium in various n o n - f u l l y the l i t e r a t u r e ,  such as F u t i a [ 1 9 7 8 ] ,  after question  the  i n f o r m i n g models i n  Grossman [1977] and Kreps  [1977].  Do p r i c e s r e v e a l i n f o r m a t i o n about p r e f e r e n c e s ? When the i n f o r m a t i o n  i s about an aggregate p r e f e r e n c e parameter  i n the u t i l i t y c l a s s e s t h a t a g g r e g a t e (extended power and l o g , and e x p o n e n t i a l ) , Chapter 3 shows t h a t the answer i s " y e s " : a fully  informing rational  there e x i s t s  expectations e q u i l i b r i u m (FRE).  t h e r e have a two p e r i o d s t a t e p r e f e r e n c e s e t t i n g , so the a p p l y to a r b i t r a r y  probability distributions.  The models  results  Introducing  i n t e r m e d i a t e p e r i o d consumption does not p r e v e n t p r i c e s from r e v e a l i n g the aggregate p r e f e r e n c e p a r a m e t e r , b u t i t  i s hard to  model the problem i n such a way t h a t agents know which p r i c e s ( o r c o m b i n a t i o n o f p r i c e s ) to i n v e r t to f i n d aggregate p r e f e r e n c e s . Introducing  random i n t e r m e d i a t e p e r i o d l a b o r income, which i s  independent o f which s t a t e o c c u r s and i s i n i t i a l l y to the agent who w i l l  earn i t ,  revealed  does not p r e v e n t p r i c e s  from  r e v e a l i n g the s a l i e n t p a r t o f aggregate p r e f e r e n c e s , as l o n g as t o t a l  ( s o c i a l ) l a b o r income i s non-random.  If  total  labor  income i s random, an e x t r a dimension o f n o i s e i s added and a FRE does not e x i s t .  However, a n o n - f u l l y  informing r a t i o n a l  expectations  125.  e q u i l i b r i u m may e x i s t , and t h i s would be an i n t e r e s t i n g problem to e x p l o r e , perhaps n u m e r i c a l l y by computer. Do p r i c e s r e v e a l i n f o r m a t i o n about the d i s t r i b u t i o n This i s a m u l t i v a r i a t e v e c t o r o f endowments.  o f a g e n t s ' endowments?  p r o b l e m , w i t h a v e c t o r o f p r i c e s and a  In a .two p e r i o d , complete market s e t t i n g where  t h e r e a r e a t l e a s t as many i n t e r m e d i a t e s t a t e s (and hence p r i c e s ) as a g e n t s , and p r o b a b i l i t i e s a r e a p p r o p r i a t e l y n o n - d e g e n e r a t e , Chapter 4 shows t h a t the answer i s g e n e r i c a l l y " y e s " , p r o v i d e d t h a t one i s o n l y i n t e r e s t e d i n the e x i s t e n c e o f a l o c a l l y f u l l y i n f o r m i n g expectations equilibrium literature,  (LFRE).  rational  Compared to o t h e r models i n  the  t h i s model makes weaker d i m e n s i o n a l i t y assumptions and  uses somewhat more c o n s t r u c t i v e a n a l y t i c t e c h n i q u e s , f o r the  price  o f g e t t i n g L F R E ' s i n s t e a d of, F R E ' s . Do p r i c e s r e v e a l and d i s t i n g u i s h between p r e f e r e n c e and parameter  probability  information?  T h i s l e a d s to a c l a s s o f m u l t i v a r i a t e  problems t h a t can  be s t u d i e d by more c o n s t r u c t i v e t e c h n i q u e s than were used i n C h a p t e r 4 , s i n c e the problems are two and t h r e e d i m e n s i o n a l . In o r d e r to model i n f o r m a t i o n about p r o b a b i l i t y  parameters,:m  Chapter 5, s p e c i f i c f a m i l i e s of p r o b a b i l i t y d i s t r i b u t i o n s normal and n o n - c e n t r a l  gamma.  are used - -  P r o b a b i l i t y parameters t h a t are s t u d i e d  a r e l o c a t i o n and l o c a t i o n - s c a l e p a r a m e t e r s .  One and two p e r i o d  models are c o n s i d e r e d where markets c o n s i s t o f bonds and a r i s k y a s s e t t h a t pays consumption d i v i d e n d s a t i n i t i a l , and f i n a l  dates.  intermediate  The one p e r i o d markets have two r e l a t i v e  prices  and c o n d i t i o n s are found under which these p r i c e s r e v e a l  preference  and p r o b a b i l i t y  relationship  information.  One c o n d i t i o n i n v o l v e s the  between bond prices (discount f a c t o r s ) and agents' common rate of impatience, where the exponential a d d i t i v e , discounted  u t i l i t y i s intertemporally  by the impatience f a c t o r .  Equivalently, this  condition involves the r e l a t i o n s h i p between current s o c i a l consumption and the un-discounted c e r t a i n t y equivalent of random next-date s o c i a l consumption.  Another condition i s the r e l a t i o n s h i p  between the value of next-date s o c i a l consumption, i f deferred  one  period, as well as various technical conditions that do not seem to be amenable to simple economic i n t e r p r e t a t i o n . In the two period models, a term structure of i n t e r e s t rates adds another (bond) p r i c e , so that aggregate r i s k preference, and p r o b a b i l i t y parameters may  be revealed by p r i c e s .  for the existence of a FRE.involve  impatience  Conditions  r e l a t i o n s h i p s l i k e those i n the  one period model, as well as the slope of the term structure of interest rates. The techniques used i n Chapter 5 are quite general and be extended to other u t i l i t y classes that aggregate, and  may  other  p r o b a b i l i t y f a m i l i e s , since the constructive a n a l y s i s involves assessing moment generating functions ( f o r exponential non-central families.  u t i l i t y ) or  moments ( f o r extended log and power) of the p r o b a b i l i t y A simple one period extension to extended log u t i l i t y  i s developed, where a FRE e x i s t s with any p r o b a b i l i t y f a m i l y , as long as the p r o b a b i l i t y parameter ranks the family i n the same order as second degree s t o c h a s t i c dominance.  Conclusion T h i s t h e s i s uses a v a r i e t y o f c o n s t r u c t i v e and t e c h n i q u e s to s t u d y the e x i s t e n c e o f a f u l l y expectations equilibrium p r i c e s convey i n f o r m a t i o n probabilities. settings.  non-constructive  informing  rational  (FRE) i n one and two p e r i o d m o d e l s , where about p r e f e r e n c e s , endowments and  Some o f the t e c h n i q u e s may be extended to  other  Bibliography  A k e r l o f , George A . 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