UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1981_A1 S53.pdf [ 5.89MB ]
Metadata
JSON: 831-1.0095184.json
JSON-LD: 831-1.0095184-ld.json
RDF/XML (Pretty): 831-1.0095184-rdf.xml
RDF/JSON: 831-1.0095184-rdf.json
Turtle: 831-1.0095184-turtle.txt
N-Triples: 831-1.0095184-rdf-ntriples.txt
Original Record: 831-1.0095184-source.json
Full Text
831-1.0095184-fulltext.txt
Citation
831-1.0095184.ris

Full Text

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 Toron to , 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 (Bus iness 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 A d m i n i s t r a t i o n ) We accept t h i s t h e s i s as conforming to the requ i red s tandard THE UNIVERSITY OF BRITISH COLUMBIA November 1980 © Gordon Ar thu r S i c k , 1980 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements fo an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary s h a l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho la r ly purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f i nanc ia l gain sha l l not be allowed without my wri t ten permission. hh3parl.mer1l.-wf Vtv^lON Of f t / j A t / c j  The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date i i A b s t r a c t Th is t h e s i s examines cond i t i ons under which p r i c e s s i g n a l i n fo rmat ion about agen ts ' p r e f e r e n c e s , endowments and /o r p r o b a b i l i t y i n f o r m a t i o n . In a m u l t i - p e r i o d economy, t h i s in fo rmat ion i s important because i t he lps agents to make i n fe rences about f u tu re p r i c e s . In a s i n g l e per iod economy, t h i s i s impor tant because, even i f agents are on ly i n t e r e s t e d in o the r agen ts ' p r o b a b i l i t y i n f o r m a t i o n , i t i s impor tant f o r them to be ab le 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 re ferences and endowments on p r i c e s . Severa l exchange economy models are cons t ruc ted to see under what cond i t i ons a f u l l y in fo rming r a t i o n a l expec ta t i ons e q u i l i b r i u m (FRE) e x i s t s i n which the r e l e v a n t i n fo rmat ion i s revea led by p r i c e s . One c l a s s o f models i s i n a two per iod s t a t e pre ference s e t t i n g i n which pre ferences e x h i b i t l i n e a r r i s k t o l e r a n c e (so t ha t aggregate pre ferences e x i s t ) . I t i s shown tha t a FRE e x i s t s t ha t r e v e a l s aggregate p re fe rence parameters. In another two per iod s t a t e p re fe rence model w i th power u t i l i t y ( in which aggregate pre ferences do not e x i s t ) , i t i s shown tha t p r i c e s g e n e r i c a l l y can revea l l o c a l i n fo rma t ion about the d i s t r i b u t i o n o f agen ts ' endowments. In another c l a s s o f models, i n both one per iod and two per iod s e t t i n g s w i th s p e c i f i c d i s t r i b u t i o n a l assumptions , (normal and non -cen t ra l gamma r e t u r n s ) , c o n d i t i o n s are found under which p r i c e s revea l i n fo rmat ion about p r o b a b i l i t i e s , aggregate r i s k pre ferences and aggregate impa t ience . i i i The t h e s i s d i scusses the no t ion o f a r a t i o n a l expec ta t i ons e q u i l i b r i u m as a s o l u t i o n of a f i x e d po in t problem. I t a l s o d i scusses i n fo rma t ion i n terms o f a -a lgebras and p a r t i t i o n s o f s t a t e spaces . i v TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES . . v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS i x Chapter 1. OVERVIEW 1 In format ion and p r i c e s 1 Overview o f the f o l l o w i n g chapters 3 2 . THE RATIONAL EXPECTATIONS CONCEPT 7 I n t roduc t i on 7 In fo rma t ion , b e l i e f s and r a t i o n a l expec ta t i ons 7 E x i s t e n c e o f r a t i o n a l expec ta t i ons e q u i l i b r i a 13 Other papers concern ing r a t i o n a l expec ta t i ons 19 Footnotes 25 3 . REVELATION OF AGGREGATE PREFERENCE PARAMETERS 28 I n t r oduc t i on 28 Nota t ion 30 A power u t i l i t y economy 33 In termedia te l a b o r income 39 FRE's w i th o the 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 c l a s s e s . . . . 44 R o l l o v e r o f p o r t f o l i o s a t date 1 46 V Chapter Page A model i n which date 0 p r i c e s revea l aggregate p re fe rences and agents r e -ba lance p o r t f o l i o s i n the in te rmed ia te pe r iod . . . 50 Conc lus ion 55 Footnotes . 56 4 . REVELATION OF INDIVIDUAL ENDOWMENTS . . . . 61 I n t r o d u c t i o n ' 61 Market s t r u c t u r e , no ta t i on and pre ferences .63 The r e v e l a t i o n o f i n fo rma t i on 68 Conc lus ion 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 Jacob ian o f l e s s than f u l l rank 83 5. PRICES REVEALING AGGREGATE RISK AVERSION, IMPATIENCE AND PROBABILITY BELIEFS 85 I n t r oduc t i on 85 No ta t ion and s e t t i n g . 89 One pe r i od models: Exponent ia l u t i l i t y 91 One p e r i o d model : Normal ly d i s t r i b u t e d re tu rns 93 One pe r iod models: Gamma d i s t r i b u t e d re tu rns . . 96 One pe r i od models : Other d i s t r i b u t i o n s and u t i l i t y c l a s s e s . . 101 Two pe r iod models : Exponent ia l u t i l i t y 104 Two pe r i od models: Normal ly d i s t r i b u t e d re tu rns . 108 Two pe r i od models: Gamma d i s t r i b u t e d re tu rns I l l Two pe r i od models: P o r t f o l i o r o l l o v e r s 118 Conc lus ion 119 Footnotes 120 v i Chapter Page 6. CONCLUDING :• REMARKS 123 What i s a r a t i o n a l expec ta t i ons e q u i l i b r i u m ? 124 Do p r i c e s revea l i n fo rmat ion about p re fe rences? 125 Do p r i c e s revea l i n fo rmat ion about the d i s t r i b u t i o n o f agen ts ' endowments? 125 Do p r i c e s revea l and d i s t i n g u i s h between pre ference and p r o b a b i l i t y parameter i n fo rmat ion? 125 Conc lus ion 127 BIBLIOGRAPHY 128 v i i LIST OF TABLES Table f ^ e 5.1 Cond i t i ons f o r a FRE i n Theorem 5.7 • • 1 1 7 j LIST OF FIGURES F igu re Page 1.1 In format ion feedback 2 2.1 A . f a m i l y o f f unc t i ons 15 3.1 Edgeworth box w i th l i n e a r c o n t r a c t c u r v e , homothet ic ! i n d i f f e r e n c e curves and endowment e 57 3.2 I n t e r i o r optimum . 59 4.1 Market regime M 65 4 .2 Market regime M' 65 4 .3 Market regime M" . . 65 5.1 S o l u t i o n o f Equat ion (5.18) 94 5.2 Graphs o f the r i g h t hand s i d e o f Equat ion (5.22) 98 5.3 Graphs of F ( e ) , the r i g h t hand s i d e o f Equat ion (5.23) 98 5.4 Graphs o f F ( e ) , the r i g h t hand s i d e o f (5.44) 113 5.5 Graphs o f G ( e ) , the r i g h t hand s i d e o f (5.45) f o r a x>cx 2 . . . . 115 5.6 Graphs o f G ( e ) , the r i g h t hand s i d e o f (5 .45) f o r a i < a 2 . . . . 115 i x ACKNOWLEDGEMENTS I would l i k e to thank the f o l l o w i n g people f o r va r ious h e l p f u l comments 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 e r r o r s , which are s o l e l y mine: Don Brown, F i s c h e r B l a c k , Michael Brennan, Erwin D iewer t , Sanford Grossman, Robert C. Mer ton, Paul Mi lgrom and Stephen Ross. Above a l l , I would l i k e to thank my a d v i s e r , A lan Kraus , f o r sugges t ing the general t o p i c cons idered he re , as we l l as f o r p rov i d i ng 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 po in t s con ta ined h e r e i n . F i n a n c i a l suppor t was prov ided by the Canada C o u n c i l , the S o c i a l Sc iences and Humanit ies Research Counc i l o f Canada, and a grant to Ya le U n i v e r s i t y by the Dean W i t t e r Foundat ion . 1. Chapter 1 Overview In format ion and p r i c e s In a dynamic economy, agents w i l l face a sequence o f markets f o r f i n a n c i a l asse t s and consumption goods, and , a t any d a t e , w i l l be unce r t a i n about what p r i c e s w i l l p r e v a i l i n those markets a t f u t u r e d a t e s . 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 uncer tan ty about exogenous parameters , such as p r o d u c t i v i t y (or the wea the r ) , war , a r r i v a l o f new techno logy , popu la t i on s h i f t s , e t c . Var ious agents o f ten have more i n fo rma t ion about these parameters than o t h e r s , or agents may s imply have d i f f e r e n t i n fo rma t ion because, f o r example, two e l e c t r o n i c s . . e x p e r t s may observe technology breakthroughs i n d i f f e r e n t product markets . At any po in t i n t ime 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 . One ca tegory r e l a t e s to what s t a t e s o f the wor ld w i l l occur i n the fu tu re ( e . g . , "severe drought i n Texas i n 1981" ) . The o ther category r e l a t e s to the parameters o f the proba-b i l i t y d i s t r i b u t i o n o f what s t a t e s o f the wor ld w i l l occur ( e . g . , "expected r a i n f a l l i n Texas i n 1981" ) . For s i m p l i c i t y , suppose tha t a l l agents have the same in fo rma t ion and b e l i e f s about what s t a tes o f the wor ld cou ld occur and tha t d i f f e r e n t i a l i n fo rma t ion pe r t a i ns on ly to d i f f e r e n t i a l i n fo rmat ion about the parameters o f the p r o b a b i l i t y d i s t r i b u t i o n ( " p r o b a b i l i t y pa ramete rs " ) . Unce r ta in t y about fu tu re p r i c e s may a l s o stem from u n c e r t a i n t y about agen ts ' p re ference parameters , such as r i s k a v e r s i o n , impat ience 2. and the functional class of agents' u t i l i t y functions (e.g., exponential, logarithmic). Also, 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 affects (future) prices. Certainly agents have better information about thei r own pre-ferences than they have about other agents' preferences. Agents'rdemands in current markets generally depend upon the information that they have about probability parameters, preferences and endowments. Thus current prices are a function of a l l of this information and hence r e f l e c t , at least to some extent, their information. Agents may then use the endogenous information impounded in prices to refine their own exogenous information. The improved information alters agents' demands and thus alters the way prices depend upon, and hence r e f l e c t , a l l of the agents' information in the f i r s t place. Information, demands, and prices are thus involved in a feedback loop, as depicted in Figure 1.1. information —-> demands >- prices : i Figure 1.1 Information feedback Because of the feedback, the market may f a i l to equilibrate. Problems of this sort are discussed in Chapter 2. The assumption that agents use a l l available information, including the endogenous information impounded in 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 called the "theory of rational expectations." As discussed in Chapter 2, this 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 shall refer to these issues as being about rational expecta-tions, in accordance with the usage in the l i t e r a t u r e . The rational expectations l i t e r a t u r e , which i s reviewed b r i e f l y in Chapter 2, has generally only dealt with cases where prices convey information about probability parameters, over-looking preference and endowment parameters, this thesis examines models where prices convey information about preferences, endow-ments, and j o i n t l y about preferences and p r o b a b i l i t i e s . Overview of the following chapters Chapter 2 provides a general discussion of information and distinguishes information from b e l i e f s . I t also defines a rational expectations equilibrium as an equilibrium i n an economy where agents use the i r own exogenous information, as well as the endo-genous information impounded i n prices. Existence of a rational expectations equilibrium i s characterized as a fixed point problem in a function space. Two types of existence problems are discussed, which motivate interest in studying the existence of f u l l y informing 4.. r a t i o n a l expec ta t i ons e q u i l i b r i a ( F R E ' s ) . The chapter conc ludes w i th a b r i e f survey o f some o f the m ic ro - and macro-economic l i t e r a t u r e on r a t i o n a l e x p e c t a t i o n s . Chapter 3 models severa l two per iod s t a t e pre ference econo- 0 mies in which aggregate pre ferences e x i s t (agents have extended power or log u t i l i t y , o r exponent ia l u t i l i t y ) . For such economies, market p r i c e s a t cu r ren t and fu tu re dates depend on an aggregate r i s k ave rs i on parameter , so tha t cu r ren t p r i c e s can r e f l e c t the va lue of the pre ference parameter , and hence he lp to r e s o l v e some o f the u n c e r t a i n t y about fu tu re p r i c e s . In such marke ts , FRE's are shown to e x i s t . The s t rong assumptions made about p re ferences and market s t r u c t u r e enable i n v e s t o r s to s i m p l i f y t h e i r i n i t i a l p o r t f o l i o c h o i c e , however, i n such a way tha t they can form opt imal demands i n the cu r ren t market , w i thou t having to assess the marke t ' s aggregate r i s k ave rs i on parameter. A model i s presented i n the Appendix to Chapter 3 i n which aggregate pre ferences 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 agen ts , f o r c i n g them to i n f e r aggregate r i s k ave rs i on from cu r ren t p r i c e s . Chapter 4 s t u d i e s the ques t ion o f whether p r i c e s can revea l i n fo rma t ion about agen ts ' endowments i n a two pe r iod w o r l d . When there are I agen ts , i n genera l there must be a t l e a s t I p r i c e s to convey the i n fo rma t ion about endowments. The problem of e s t a b l i s h -ing the ex i s t ence o f FRE 's t ha t convey m u l t i v a r i a t e i n fo rma t ion i s more d i f f i c u l t than tha t o f a n a l y s i n g the u n i v a r i a t e FRE's o f Chap-te r 3. 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. gener i c e x i s t e n c e o n l y . That i s , i f a FRE does not e x i s t f o r an economy, a "smal l p e r t u r b a t i o n " o f the economy y i e l d s a FRE, and , i f an economy has a FRE, a l l " s u f f i c i e n t l y c l o s e " economies have F R E ' s . The gener i c ex i s t ence o f FRE 1 s i n one-pe r iod economies, where p r o b a b i l i t y i n fo rma t ion 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 rong d i m e n s i o n a l i t y assumptions by A l l e n [1978, 1979] , She uses d i f f e r e n t i a l topo logy to get these r e s u l t s . The r e s u l t s i n Chapter 4 s tudy the e x i s t e n c e o f l o c a l l y f u l l y in fo rming r a t i o n a l expec ta t i ons e q u i l i b r i a (LFRE 's ) o n l y : i f the vec to r o f endowments i s con f ined to a s u i t a b l y smal l s e t , p r i c e s revea l the endowment vec to r i n e q u i l i b r i u m . By on ly search ing f o r L F R E ' s , Chapter 4 e s t a b l i s h e s gener i c ex i s t ence under s l i g h t l y weaker d i -mens iona l i t y assumptions than A l l e n , by us ing on ly wel l -known a n a l y t i c r e s u l t s about i m p l i c i t f u n c t i o n s . In Chapter 5 , one and two pe r iod economies w i t h one r i s k y asse t are s tud ied to see when p r i c e s can revea l i n fo rma t ion about p r o b a b i l i t y parameters and pre ference parameters (aggregate r i s k a v e r s i o n , and, i n the two per iod economy, impa t i ence ) . In c o n t r a s t 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 i n fo rma t ion r e s u l t s are c o n s t r u c t i v e . The r e s u l t s are s t ronger than gene r i c r e s u l t s : they g ive s u f f i c i e n t , and i n some c a s e s , necessary c o n d i t i o n s f o r a FRE to e x i s t . The models are based upon i n t e r t e m p o r a l l y a d d i t i v e exponent ia l u t i l i t y f unc t i ons and i n v o l v e two s p e c i f i c p r o b a b i l i t y f a m i l i e s f o r the increments to the s o c i a l weal th p rocess : normal 6. and non -cen t ra l gamma. 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 t y p i c a l l y i nvo l ve the r e l a t i o n s h i p between agen ts ' r a t e o f impat ience and d i s c o u n t bond p r i c e s ( i n the one pe r iod models) and the s lope of the term s t r u c t u r e o f i n t e r e s t ra tes ( i n the two pe r i od mode ls ) . Chapter 6 p rov ides some conc lud ing remarks. 7. Chapter 2 The Rational Expectations Concept Introduction This chapter provides a general discussion of rational expec-tations models, in terms of the i r d e f i n i t i o n , existence and general properties. I t starts with a general discussion of information, in terms of a-algebras and partitions in probability spaces, and pro-ceeds to a characterization of rational expectations e q u i l i b r i a as fixed points (in a Banach space). Continuity and measurability problems that create problems for the existence of rational expec-tations 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 rational expectations e q u i l i b r i a (FRE's), the existence of which i s more readily v e r i f i a b l e than the existence of general rational expectations e q u i l i b r i a . The chapter concludes with a survey of some of the rational ex-pectations models in the l i t e r a t u r e . Information, b e l i e f s and rational expectations Consider a market system for S goods involving uncertainty about, say, future endogenous or exogenous production l e v e l s . Suppose agent i ( i = l , I) receives information.A about the true value of the un-certain variables. One may think of this in terms of a probability space^ (ft, 8, p ), with state space a, a-algebra B and probability measure P^. Agent i's be 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 take conditional 2) expectations. • 8. I f the s t a t e space a has f i n i t e l y many elements ( s t a t e s ) , B i s the c o l l e c t i o n o f a l l subsets o f Q (B=2 f i ) , and P.. i s generated by the non-negat ive p r o b a b i l i t y o f each s t a t e . Agent i ' s in fo rma-t i o n A.j may be regarded as a p a r t i t i o n o f o, ( a c t u a l l y the p a r t i t i o n generates A..) such tha 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 , but 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 . For example, i f n = {1 n} , suppose A., i s generated by the p a r t i t i o n {{1,2>, {3} , { 4 , . . . , n } } . Then, i f s t a t e 1 o c c u r s , agent i on ly knows tha t e i t h e r s t a t e 1 or s t a t e 2 o c c u r r e d . C o n d i t i o n i n g on a p a r t i t i o n member B i n v o l v e s forming the usual c o n d i t i o n a l p r o b a b i l i t i e s . P ( t j } | B ) = g P ( j p j k ) ( j e B ) . keB The most impor tant c o n d i t i o n a l expec ta t i ons are those used to generate demand f u n c t i o n s . Le t A CE+ be a f e a s i b l e se t o f p r i c e s , where £ + i s the non-negat ive o r than t i n Euc l i dean n -space . Le t (co) e be agent i s endowment a t co e n . The - excess demand f u n c t i o n : ^ :Axnx{A.C2 N|A. i s a s u b - a - a l g e b r a o f B} -> £ S s o l v e s , f o r each mtt and i n fo rma t i on A^ the expected u t i l i t y max imiza t ion problem. That i s , (2 .1) ^ ( p ^ . A . ) = arg max E ^ U ^ x + x ^ c o ) ) ^ . ) . {x e & s | p' x=0} Here, "a rg max" r e f e r s to the maximiz ing argument o f a max imiza t ion 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 on ly 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 f x\(to) i s a non-cons tan t f u n c t i o n o f u , we r e q u i r e i t to be an A . -measurab le random v a r i a b l e , so tha t agent i knows h i s budget c o n s t r a i n t . I f c u t i l i t y i s s u f f i c i e n t l y smooth, e t c . then C j (p ,» ,A . . ) : Q, •> & w i l l be a proper (A^-measurable) random v a r i a b l e f o r a l l p, A, . . I f n i s a f i n i t e s t a t e space , t h i s means tha t ( p , . / 5 A . ) w i l l be cons tan t over members o f the p a r t i t i o n genera t ing A. . , s i n c e agent i cannot 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 p a r t i t i o n genera t i ng A ^ . Loose ly 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 p r o b a b i l i t y s t a t e space Q, agent i r e c e i v e s some (but perhaps not a l l ) i n fo rma t ion about the va lue o f co (he can t e l l whether o r not us i s i n any g iven event i n A.., but not whether to i s o r i s not i n any s m a l l e r e v e n t s ) , as we l l as the va lue o f h i s endowment. He then so l ves (2.1) f o r h i s (random) excess demand f u n c t i o n ^ ( 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 (2.2); z. t ; i (p, to,A 1 . ) = 0 . This d e f i n e s the p r i c e p, assuming, f o r s i m p l i c i t y , t ha t e q u i l i b r i u m p r i c e s a re un ique . Set P(W) = p. Repeat t h i s f o r a l l co to ge t the market c l e a r i n g p r i c e f u n c t i o n p". Under app rop r i a te r e g u l a r i t y c o n d i -t i o n s , p w i l l be a 8 -measurable f u n c t i o n o f o> and hence a t rue random v a r i a b l e . Indeed, s i nce p can on ly vary accord ing to the i n fo rma t ion { A . . } communicated to the agen ts , p should be A -measurable where A = A-^vA^v.. .vAj i s the s m a l l e s t a -a l geb ra c o n t a i n i n g a l l the A . j ' s . I f i s f i n i t e , A corresponds to the common re f inement of a l l the p a r t i t i o n s genera t ing the A . j ' s . In some sense , A i s the " s o c i a l endowment" o f i n f o r m a t i o n . The impor tant po in t i s t ha t p = P(OJ) v a r i e s w i th the i n f o r -mation tha t (o ther ) i n d i v i d u a l s r e c e i v e . A r a t i o n a l i n d i v i d u a l w i l l i n c l u d e the i n fo rma t i on conveyed by P i n s e l e c t i n g h i s demand f u n c t i o n . Le t P - 1(A) denote the a - a l g e b r a generated byp (as de f i ned i n foo tno te 2 . , i . e . , the i n fo rma t i on conveyed by the p r i c e f u n c t i o n Let 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 ) , i . e . , a l l the i n fo rma t i on con ta ined i n A^ and P. Th is i s a l l the i n fo rma t i on a v a i l a b l e to agent i . ' Th is 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 r a t i o n a l i n d i v i d u a l w i l l now have f o r each co, excess demand a t p r i c e p o f 5^ (p^.A^vP - 1 ( A ) ) . Note tha t now, the random excess de-c mand 5^  depends on the whole p r i c e f u n c t i o n P : fi-*- & + , s i n c e . i t communicates i n f o r m a t i o n , as we l l as on the p r i c e p, s i n c e i t de te r -mines the budget c o n s t r a i n t . A l s o , £ . depends on co on ly through the i n fo rma t i on conveyed by A ^ v p ' ^ A ) , so t ha t under app rop r i a te regu - ; l a r i t y c o n d i t i o n s £ . ( p , •,A^ v p ~ ^(A) ) i s A.vp~^(A ) -measurab le . Now, f i x co and impose the market c l e a r i n g c o n d i t i o n ( 2 . 3 ) Z .5 . ( P , (o,A . vp ( A ) ) = 0 -\ 11. Set p'(co) = p. Repeat ing t h i s f o r a l l coeft de f i nes a new p r i c e f u n c t i o n p1 :R A wh ich , under app rop r i a te r e g u l a r i t y c o n d i t i o n s w i l l be A -measurab le ( i . e . , random). Of c o u r s e , s i n c e agents have formula ted t h e i r demands w i th r espec t to a new i n fo rma t i on s e t ( A ^ v p " ^ ( A ) , r a t h e r than A . . ) , the new p r i c e f u n c t i o n p' w i l l , i n g e n e r a l , be d i f f e r e n t from p. I n d i v i d u a l s must now r e - r e v i s e t h e i r expec ta t i ons us ing A v p ' ^ ( A . ) r a t h e r than A . i n ( 2 . 1 ) . With the market c l e a r i n g c o n d i t i o n , t h i s g i ves a new p r i c e f u n c t i o n p", and so on ad i n f i n i t u m . The ques t ion o f whether such an analogue of the tatonnement process 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 not be ana lysed he re . The impor tant po i n t i s t ha t a f i x e d po in t ( i f i t e x i t s ) o f such a sequence of re -ad jus tments i s c a l l e d a r a t i o n a l expec ta t i ons e q u i l i b r i u m . A r a t i o n a l expec ta t i ons e q u i l i b r i u m occurs when there e x i s t s an A -measurab le p r i c e f u n c t i o n (random v a r i a b l e ) p : n--:-* & + such tha t i f excess demands are de f i ned by (2 .1 ) then (2 .4) s i ? i ( P (o J ) ,co,A i vp" 1 (A)) = 0 . (For mathematical 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 "a lmost s u r e l y . " ) Thus , agent i uses h i s own exogenous i n fo rma t ion A., and the endogenous i n fo rma t ion 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 ha t the p r i c e f u n c t i o n p which agents assume holds i n f o rmu la t i ng t h e i r expec ta t i ons and demands i s a c t u a l l y the one tha t c l e a r s the market . Th is i s the " s e l f - f u l f i l l i n g " nature o f a r a t i o n a l expec ta t i ons e q i l i b r i u m . Thus, the cons i s tency tha t 12. i s i m p l i c i t i n 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 o f b e l i e f s o r e x p e c t a t i o n s , f o r there cou ld be d i f f e r e n t i a l b e l i e f s . (Pn- d i f f e r e n t from P^) o r , even i n c o r r e c t b e l i e f s (P^ d i f f e r e n t from N a t u r e ' s " t r u e " p r o b a b i l i t y law P ) . Ra t i ona l expec ta t i ons r e a l l y on l y imposes a cons i s tency on agen ts ' i n fo rma t ion A^VP"^(A). Rat iona l expec ta t i ons r e q u i r e t ha t agents on ly f o r e c a s t the p o s s i -b i l i t y o f f e a s i b l e events — tha t i s , . t h e y - u s e the s t a te space ft, and have a a -a lgeb ra A..VP~^ (A;). C B. I t a l s o r equ i r es t ha t agents c o r -r e c t l y use a l l a v a i l a b l e i n fo rma t i on (as i n A- and p ~ ^ ( A ) ) . B u t , i t does not r e q u i r e any homogeneity o r co r rec tness o f 1 p r o b a b i l i t y b e l i e f s P.j, which a re exogenous t o . t h e model. In a repeated economy, agents may update P.. i n a Bayes ian manner, and over t ime , P^ may converge to n a t u r e ' s " t r u e " P. To make r a t i o n a l expec ta t i ons models y i e l d e m p i r i c a l l y t e s t a b l e i m p l i c a t i o n s , i t o f t en seems necessary to add the hypo thes is t ha t b e l i e f s are homogeneous and " c o r r e c t " ( c o i n c i d e n t w i th P ) . Moreover , i f agents have d i f f e r e n t i a l b e l i e f s , they must a d j u s t f o r t h i s i n order to i n f e r i n fo rma t i on from o ther agen ts ' a c t i o n s o r from p r i c e s . For example, suppose a s u p e r i o r s tock a n a l y s t ( c h a r a c t e r i z e d by a l a rge s - a l g e b r a A . , o r a very r e f i n e d p a r t i t i o n ) i s a l s o an i n c u r a b l e o p t i m i s t and can be on l y observed to be "buy ing s tock " or "buy ing l o t s o f s t o c k . " A l e s s w e l l - i n f o r m e d agent , but w i th 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 , w i l l f i n d i t opt imal to s e l l s tocks when the a n a l y s t merely buys, and, to buy s tocks when the a n a l y s t buys l o t s o f s t o c k . 13. C l e a r l y , agents may have e i t h e r homogeneous or d i f f e r e n t i a l b e l i e f s w h i l e having e i t h e r homogeneous or d i f f e r e n t i a l i n f o r m a t i o n . Ra t i ona l expec ta t i ons i s a misnomer f o r t h i s theory i n t ha t i t i s r e a l l y on l y a theory 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 o f b e l i e f s o r e x p e c t a t i o n s . 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 expec ta t i ons e q u i l i b r i a w i th 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 ha 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 ten makes c losed form s o l u t i o n s imposs ib le to a t t a i n . Hetero-geneous b e l i e f s merely c loud the problem o f s tudy ing d i f f e r e n t i a l i n f o r m a t i o n . Ex i s t ence o f r a t i o n a l expec ta t i ons e q u i l i b r i a 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 d i f f i -c u l t i e s a s s o c i a t e d w i th e s t a b l i s h i n g the e x i s t e n c e of a r a t i o n a l expec ta t i ons 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 s t a b i l i t y or dynamics. In s tandard models o f u n c e r t a i n t y w i thou t d i f f e r e n t i a l i n f o r m a t i o n , we have . A . , . = A( \ / i ) (and A = B i n the c e r t a i n t y c a s e ) , so t ha t the p r i c e f u n c t i o n conveys no i n fo rma t ion ( i . e . , A^vp~^(A) = A^, = A ( \ / i ) ) . Ex i s tence of 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 tandard methods, such as i n Debreu (1959) or Arrow and Hahn (1971) . In t h i s c a s e , p r i c e p i s a f i x e d po in t o f a cont inuous mapping o f A i n t o i t s e l f , w i t h the proper ty tha t a f i x e d po in t corresponds to a po in t o f zero aggregate excess demand. For the general r a t i o n a l expec ta t i ons problem w i th d i f f e r e n t i a l 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 po 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 po in t i n Euc l idean S -space . Th is f u n c t i o n space , endowed w i th an app rop r i a te topo logy , w i l l be a Banach space . T h i s , i n i t s e l f , i s no prob lem, f o r there are f i x e d po in t theorems f o r Banach spaces . The problem i s tha t i t may not be p o s s i b l e to pos tu l a te enough c o n t i n u i t y , compactness and o ther topo-l o g i c a l p r o p e r t i e s on the exogenous pa r t s o f the problem to ensure c o n t i n u i t y o f any use fu l mapping i n the Banach space o f p r i c e random v a r i a b l e s . Th is occurs because the excess demand random v a r i a b l e s S.j(p, *,A.jVP ^ ( A ) ) are no t , i n g e n e r a l , cont inuous f u n c t i o n s , under any a p p r o p r i a t e t opo logy , o f the p r i c e random v a r i a b l e P. That i s , i n g e n e r a l , the i n fo rma t i on communicated by p r i c e s , P~^(A) i s no t , i n some sense , a cont inuous 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}, parameter ized by 6e [0,1], and de f ined by / 6 l -Q Suppose t ha t agents are a t tempt ing to l e a r n something about ~<o by obse rv ing P g ^ ) . The i n v e r t i b i l i t y o f p 0 depends on i t s monoton ic i t y and hence on the zeros o f i t s d e r i v a t i v e s . There are fou r c a s e s : 1. 6e [0,1 /5) . There are no s t a t i o n a r y po in t s f o r P e(w) f o r u>. 0 . 2. e = 1 /5 . Pg M = 0 on ly when «D = 0 . 3 . 0 £ ( 1 / 5 , 1 /2 ) . P0'H = 0 f o r e x a c t l y one .u* > 0 , and °° as o + 1 /2 " . 4 . e e [ 1 / 2 , 1 ] . p ' (co) * 0 f o r a l l co > 0. This i s graphed i n F igure 2 . 1 . For 0>. ^ ,P9 (w) i s an i n v e r t i b l e f u n c t i o n o f to, but f o r 6 < i t i s not i n v e r t i b l e . C a l l u i and u2 confounding i f p f l ( u i ) = Pfl(to2) = P, s a y . S ince the s t a t i o n a r y po i n t w o f p [{-) goes to + <>° as e -> 1 / 2 " , there are a r b i t r a r i l y l a r g e confounding p a i r s c o l s u>2 w i th Ico^-u) 2|' a r b i t r a r i l y l a rge and wi th p^(to{-) a r b i t r a r i l y c l o s e to 1. I f an a g e n t ' s p re fe rences are markedly \ d i f f e r e n t i n s t a t e s tn1 and c o 2 , the numerical va lue o f the excess demand w i l l be very v o l a t i l e and d i scon t i nuous a t e = \ as p Q suddenly becomes i n v e r t i b l e . Th is i n fo rma t i on d i s c o n t i n u i t y a t 6 = j | i s not removable and g e n e r a l l y leads to 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 f o i l the a p p l i c a t i o n o f f i x e d po i n t theorems to e s t a b l i s h the e x i s t e n c e of e q u i l i b r i u m . In a d d i t i o n to these c o n t i n u i t y prob lems, there are o ther problems (which may be termed " m e a s u r a b i l i t y problems") w i th e s t a b l i s h i n g the e x i s t e n c e of a r a t i o n a l expec ta t i ons e q u i l i b r i u m . We have a l r e a d y argued tha t agent i ' s excess demands w i l l be measurable w i th respec t to h i s i n fo rma t ion A^VP"^(A); t ha t i s , h i s demands w i l l vary on ly accord ing to the in fo rma t ion he a c t u -a l l y r e c e i v e s . On the o ther hand, d i f f e r e n t p ieces o f i n fo rma t ion 17. w i l l g e n e r a l l y l ead to d i f f e r e n t demands, so tha t h i s excess demand f u n c t i o n w i l l o f t en generate the whole a - a l g e b r a >A.jVP~^ ( A ) . For a more concre te i n t e r p r e t a t i o n , suppose the s t a t e space ^ i s f i n i t e , so tha t we need on ly c o n s i d e r the p a r t i t i o n s tha t generate the i n fo rma t i on a - a l g e b r a s . (A f i n e r p a r t i t i o n corresponds to s u p e r i o r i n f o r m a t i o n , e t c . ) With an obvious change of n o t a t i o n , l e t A^ be agent i ' s exogenous in fo rma t ion p a r t i t i o n , so t ha t he can d i s t i n g u i s h s t a t e s toeft on ly i f they are i n d i f f e r e n t members o f h i s p a r t i t i o n . Le t p~^(A) be the p a r t i t i o n generated by p i . e . , the p a r t i t i o n c o n s i s t i n g o f equ iva lence sets o f s t a t es co which map to the same p r i c e under P. Then A.VP'^A) i s the common re f inement o f A^ and P~^(A). To say tha t agent i ' s excess demand f u n c t i o n i s A jVP ^ (A) -measurab le means tha t i t i s cons tan t on members o f the p a r t i t i o n . In g e n e r a l , excess demands w i l l be d i f f e r e n t on d i f f e r e n t members o f the p a r t i t i o n ( i . e . , w i l l generate the whole p a r t i t i o n ) . The e q u i l i b r i u m c o n d i t i o n i s t ha t excess demands sum to z e r o , f o r a l l coeft . But excess demands t ha t vary on d i f f e r e n t p a r t i t i o n s (or a -a l geb ras ) w i l l no t , i n g e n e r a l , sum to a cons tan t f u n c t i o n . For example, 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 (or s i m p l y , d i f f e r e n t ) p a r t i t i o n than the o t h e r , because such excess demand cannot sum to zero f o r a l l coefi.. I f we can r u l e out d i f f e r e n t i a l i n fo rma t ion ( a f t e r t ak ing i n t o account endogenous p r i c e i n fo rma t ion ) then we may get an e q u i l i b r i u m . That i s , , we r e q u i r e , of.; an^e.quit-14.brium p that vp~'(A)..= AjVp~'(A) for a l l i, j . Hence, we have A^p'^A) c A C (A 1V p" 1(A))V...V(A IVP" 1(A))= A.VP _ 1(A), so that A = A^vp - 1(A) for a l l i . That is i n equilibrium, a l l 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 this also tends to imply that P""'(A) =A, that i s , a l l information i s revealed by prices. I f a rational expectations equilibrium exists with P _ 1 ( A ) = A , we shall say that a f u ! l y  informing rational expectations equilibrium (FRE) e x i t s . This measurability argument suggests that, 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_1 (A))., agents must be f u l l y informed for equilibrium 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 of to, a FRE continues to e x i s t , there are no information discontinuities so that the continuity and measurability problems are solved simultaneously with FRE's. In the following chapters we study the existence of FRE's only. There i s a straightforward procedure for verifying the existence of a FRE. One simply solves for the fully-informed demand functions 5 . j(p,<",A). The market clearing condition then defines P, from which i t i s then, in p r i n c i p l e , possible to check whether A =. p " ^ ( A ) . in practice, there may be simple parameters which are s u f f i c i e n t for defining conditional 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 independent , i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s . A s u f f i c i e n t 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 ha t p be an i n v e r t i b l e f u n c t i o n o f these parameters . Th is i s the technique used i n the f o l l o w i n g c h a p t e r s . Other papers concern ing r a t i o n a l expec ta t i ons The term " r a t i o n a l e x p e c t a t i o n s " was co ined i n a paper by Muth [1961] . H is model was the f i r s t o f a s e r i e s o f macroeconomic models developed by va r ious au tho rs . These models g e n e r a l l y have homogeneous in fo rma t ion .(A.= A, V i ) and homogeneous b e l i e f s (P..= P , \ / i ) . To enhance the e m p i r i c a l t e s t a b i l i t y o f these models, i t i s u s u a l l y assumed t ha t P a l s o represents Na tu re ' s " t r u e " p r o b a b i l i t y law. so t ha t ex post p r o b a b i l i t y laws are the same as ex ante p r o b a b i l i t y l aws . In these models , the assumption o f r a t i o n a l expec ta t i ons i s b a s i c a l l y a s t rong cons i s t ency c r i t e r i o n tha t i s used to ob ta i n s t ronge r conc lus i ons i n macro models. For a good rev iew o f t h i s l i t e r a t u r e , see S h i l l e r [1978] , In f i n a n c e , the homogeneous b e l i e f s and i n fo rma t i on assumption has been used by Cox, I n g e r s o l l and 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 r a t e s . In a genera 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 process o f i n t e r e s t r a t e s , they used the assumption o f homogeneous and c o r r e c t b e l i e f s to r e l a t e the i n t e r e s t r a t e process tha t agents assume ho ld when f o rmu la t i ng demands to the process t ha t a c t u a l l y r e s u l t s , g iven agen ts ' demands. The micro-economic and general e q u i l i b r i u m a p p l i c a t i o n s o f r a t i o n a l expec ta t i ons 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 -t i a l i n f o rma t i on and homogeneous (and c o r r e c t ) b e l i e f s . Here the c e n t r a l ques t ion i s o f ten 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 i n fo rma t i on r a t i o n a l expec-t a t i o n s model i s the " lemons" model o f used ca r markets by A k e r l o f [1970] . In t h i s market , the s e l l e r has b e t t e r i n fo rma t ion about the q u a l i t y o f h i s ca r than has the buyer . In the absence o f some method o f s i g n a l l i n g q u a l i t y , the owner o f a good c a r w i l l on ly r e c e i v e a p r i c e cor respond ing to an average ( i . e . , i n f e r i o r ) c a r . He withdraws h i s c a r from the market , l e a v i n g a lower average q u a l i t y , and,' i n a l i k e manner, the owners o f the next grade o f ca rs leave the market , f i n a l l y l e a v i n g on ly the lowest q u a l i t y " lemons" on the market . In t h i s c a s e , i n fo rma t ion i s conveyed by the f a c t tha t a c a r - i s o f f e r e d on the market ( f o r a g iven p r i c e ) . At any g iven p r i c e , there i s p o s i t i v e supply and p o s i t i v e demand, but the two are never e q u a l , so the market f a i l s . In f i n a n c e , there are 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, 1978a] -and-of 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 Heinke l [1978] . These models use d i f f e r e n t i a l i n fo rma t ion to e x p l a i n va r i ous e m p i r i c a l l y observed phenomena tha t h i t h e r t o cou ld on ly be exp la i ned by i n s t i t u t i o n a l r i g i d i t i e s l i k e t r a n s a c t i o n s c o s t s . The c a p i t a l s t r u c t u r e models show how i t i s p o s s i b l e f o r a f i r m to have an opt imal c a p i t a l s t r u c t u r e i n a t a x - f r e e w o r l d , i n 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 model , the p ropo r t i on o f equ i t y f i nanced by i n s i d e r s s i g n a l s i n f o rma t i on about the random re tu rns on c a p i t a l inves tment . Th is r e s u l t s i n a f u l l y in fo rming r a t i o n a l expec ta t ions e q u i l i b r i u m i n which ent repreneurs 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 opt imal investment would be i f a l l agents exogeneously have the same i n f o r m a t i o n . Th is occurs because, w i th d i f f e r e n t i a l i n -fo rmat ion the b e n e f i t s o f i n s i d e investment i n the f i r m are not on l y the fu tu re "returns, but a l s o the c u r r e n t re tu rns o f s e l l i n g pa r t o f the f i r m a t the h igher p r i c e tha t r e s u l t s when i t i s c l a s s i f i e d as a h i g h - r e t u r n f i r m . Th is d i s t o r t i o n i s due to the d i f f e r e n t i a l 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 behav iou r . In the Ross models , managers have s u p e r i o r i n fo rma t ion about f i r m type and have compensation based i n c e n t i v e schemes t ha t a l l ow the market to pe rce i ve f i r m type by observ ing the manager's compen-s a t i o n f o rmu la . In seve ra l o f the Ross models , the compensation scheme i s based on the cu r ren t va lue o f the f i r m , as we l l as the f u tu re va lue o f the e q u i t y . By va ry i ng the f i r m ' s debt l e v e l , managers t a i l o r t h e i r compensation scheme to s i g n a l f i r m t ype , y i e l d i n g an opt imal debt l e v e l . The Heinke l model uses a bor rower ' s cho i ce 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 fo rma t i on about how r i s k y h i s p r o j e c t i s . R i sky borrowers would l i k e to borrow a l a r g e amount, s i n c e , w i th l i m i t e d 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 igh p r o j e c t r e t u r n s , but are not pena l i zed s i g n i f i c a n t l y f o r low r e t u r n s . 22. By r a t i o n i n g borrowers ( i . e . , keeping the face va lue o f debt l o w ) , the l i m i t e d l i a b i l i t y f ea tu re i s o f l e s s va lue o f r i s k y bor rowers , and lenders w i l l on l y lend to low r i s k bor rowers . Thus, the borrowers revea l t h e i r t rue type ( " s e l f - s e l e c t " ) by choosing d i s -t i n c t i n t e r e s t / l o a n s i z e p a i r s . However, l o w - r i s k borrowers are r a t i oned to a s m a l l e r loan s i z e than they would choose i f r i s k c l a s s e s were exogenously r e v e a l e d . 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 "honest " versus " d i s h o n e s t " bor rowers . These models o f debt markets and o f c a p i t a l s t r u c t u r e , as we l l as r e l a t e d models o f i nsu rance markets by R o t h s c h i l d - S t i g l i t z [1976] have the f ea tu re t h a t , a l though agents are p r i c e - t a k e r s , informed agen t s ' a c t i o n s a f f e c t the " m a r k e t ' s " pe rcep t ions o f the goods they are o f f e r i n g ( e . g . , " f i r m t y p e " ) , and hence the l i t e r a -tu re cen te rs on i n c e n t i v e schemes, moral h a z a r d , agency and i n f o r -mation s i g n a l l i n g behav iour . There are o ther models , however, i n which agents do not ac t i n ways to e x p l i c i t l y encourage o r d iscourage the d i ssem ina t i on o f t h e i r p r i v a t e i n f o rma t i on to the market . For example, Grossman [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 in fo rma-t i o n about the n e x t - p e r i o d va lue o f a r i s k y s e c u r i t y . The c u r r e n t p r i c e i s determined by the aggregate con ten t o f tha t i n fo rma t i on (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 i n f o r m a t i o n ) . Moreover, the r e l e v a n t aggregate i n fo rma t ion can be i n f e r r e d by observ ing p r i c e a l o n e . Th is leads to one d i s t u r b i n g f ea tu re o f Grossman's model , 23. as po in ted out by Grossman [1976, p . 582 ] , Kraus [1976] and Grossman-S t i g l i t z [1976] . I f the p r i v a t e c o l l e c t i o n o f i n fo rma t i on i s c o s t l y , but the obse rva t i on o f p r i c e s i s c o s t l e s s , agents w i l l not c o l l e c t p r i v a t e i n f o r m a t i o n , expec t ing tha t i t w i l l be revea led i n p r i c e s , wh i ch , o f c o u r s e , i t w i l l no t , i f nobody c o l l e c t s i n f o r m a t i o n . 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 . There are models by Green [1977] , R o t h s c h i l d [1976 ] , and.Jordan-Radner [1977] i n which e q u i l i b r i u m f a i l s to o b t a i n i n r a t i o n a l expec ta t i ons models. For s i m p l i c i t y , these models deal w i th two ( c l a s s e s o f ) agen ts : in formed and uninformed. In a f i n i t e s t a t e - s p a c e model , the informed agents can d i s t i n g u i s h s t a tes i n a more r e f i n e d p a r t i t i o n than the uninformed agen ts . However, p r o b a b i l i t y b e l i e f s are homogeneous i n the sense tha t both agents agree on the p r o b a b i l i t i e s o f events i n the c o a r s e r , uninformed p a r t i t i o n . In these models , the uninformed agents e i t h e r become informed o r not when they observe e q u i l i b r i u m p r i c e s . I f they a re i n fo rmed , t h e i r demands are such tha t p r i c e s are not f u l l y i n f o r m i n g , but i f they are not in fo rmed, on ly f u l l y i n fo rm ing p r i c e s can e q u i l i b r a t e the economy, s i nce informed agen ts ' excess demands are d i f f e r e n t i n each member i n the r e f i n e d p a r t i t i o n . Thus, m e a s u r a b i l i t y problems fo r ce the non -ex i s tence o f a r a t i o n a l expec ta t i ons e q u i l i b r i u m . In the Jordan-Radner model , non -ex i s tence occurs even under pe r tu rba t i ons o f the unde r l y i ng parameters o f the economy — tha t i s , - n o n - e x i s t e n c e i s , i n a sense , g e n e r i c . On the o ther hand, there are gener i c e x i s t e n c e r e s u l t s f o r FRE's by Radner [1978] and A l l e n [1978, 1979] , which are based on theorems i n d i f f e r e n t i a l topo logy . The general r e s u l t i s t ha t the dimension 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 ) exceed the dimension o f the i n fo rma t ion space . These d i m e n s i o n a l i t y r e s t r i c t i o n s are v i o l a t e d i n the Jordan-Radner counterexample. These r e s u l t s have a l l addressed the communication o f p r o b a b i l i t y i n fo rma t ion o n l y . In format ion about p re ferences and endowments i s a l s o impor tant and the ques t ion o f whether o r not t h i s can be con-veyed by p r i c e s i s d i scussed i n K raus -S i ck [1979a, 1979b, 1980] . The f o l l o w i n g chapters d i s c u s s those r e s u l t s as we l l as ex tens ions o f those r e s u l t s . .25. Footnotes to Chapter 2 1. A r igourous understanding o f measure - theo re t i c p r o b a b i l i t y theory i s not r equ i r ed to understand the t h r u s t o f the argument presented here . The theory 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 . I I , Chapters IV and V] and re fe rences t h e r e i n . Roughly s p e a k i n g , s e l e c t i o n o f a s t a t e u>eo, corresponds to a random r e a l i z a t i o n o f the system. The o -a l geb ra 8 i s a f a m i l y o f subsets ( c a l l e d events) 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 ] ass igns to each event B e B a non-negat ive p r o b a b i l i t y 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 any subset o f n t ha t i s not an event i s unde f ined . A random v a r i a b l e i s a f u n c t i o n n : o, -> & n ( f o r some n) which i s B-measurab le . That i s , f o r any ( a - ! , . . . , a n ) e & n ' {co e ft | n . (co) < _ a . , j = l , . . . , h } e B , J J J.L. where n.(oo) i s the j component o f n (u>). Roughly s p e a k i n g , t h i s 3 means tha t n does not vary too much - - any change i n the va lue o f rt can be captured by an event i n B . 2 . For example, l e t n : & & n be a random v a r i a b l e . Then ( see , f o r example, F e l l e r [1966, V o l . I I , pp. 160-162 ] ) , the c o n d i t i o n a l  expec ta t i on E.(nU^) i s the A.j-measurable random v a r i a b l e 5 : n -»- &n :such that /E.Q^^JAeCwOdPiU) = J A n ( c J ) d P i ( u ) = E . ( l A n ) f o r a l l A e A . where 1„ : o, -»- &. i s the se t i n d i c a t o r f u n c t i o n de f i ned by 1 i f co e A 0 i f to f. A Note that, since F..,-(ji | Aj) i s a random variable i t depends on the r e a l i z a t i o n coeft and should r e a l l y be written as, say (n| Aj . Following conventional usage, we suppress the e x p l i c i t references to C J . I f probability densities e x i s t , or probability i s discrete, this i s i n accordance with the usual notion of conditional expectation or probability. Also, i f AeB i s an event, define = {<t>,A,ft\A,ft}. Since (n| ) i s A.-measurable, i t i s constant on the event A and the constant i s unique i f A has positive probability. Let E.(n|A) be that constant value on A. This conforms with our conventional notion of the non-random conditional expectation given an event of positive probability. To condition on a random variable, p say, i s to condition 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,... ,an) § &n. I f ft i 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 of ft generated by inverse images of points i n £ n under p. The p a r t i t i o n 1 S {{co : p(co) = p} : .pe&n } The a-algebra generated by p distinguishes only those points which are mapped to diff e r e n t points by p, and hence i s the information conveyed by observing p. This i s not to suggest t ha t n o n - f u l l y in fo rming r a t i o n a l expec ta t i ons e q u i l i b r i a do not e x i s t . I f there are two dimensions o f u n c e r t a i n t y (a =•£. ) but an informed agen t ' s demand and hence p r i c e conveys on ly a u n i v a r i a t e f u n c t i o n of the u n c e r t a i n t y , i t 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 e x i s t . Th is o c c u r s , f o r example, i n Grossman [1977] , i n which ran -dom supply i s added to unce r t a i n t y about the va lue o f a r i s k y a s s e t . The informed agen ts ' demands are u n i v a r i a t e i n nature and generate a "one-d imens iona l a - a l g e b r a " ( i . e . , a a - a l g e b r a generated by one-d imensional t r a n s l a t i o n s of a m a n i f o l d , so tha t there i s on ly one remain ing degree o f freedom on which v a r i a t i o n can occur i n a ) . In h i s model , opt imal demands f o r the uninformed agent are merely c o n d i -t i o n a l expec ta t i ons o f opt imal in formal demands, and hence can generate the same one d imensional © -a lgeb ra , a l l o w i n g excess demands to sum to z e r o . I f 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 agen ts , o r have some in fo rma t ion tha t " in fo rmed" agents do not have, then un in -formed demands may not generate the same a - a l g e b r a as informed agen ts , or have some in fo rma t ion tha t " in fo rmed" agents do not have, then uninformed demands may not 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 prevents the e x i s t e n c e of e q u i l i b r i u m . I t would be i n t e r e s t i n g to study the robustness o f t h i s n o n - f u l l y in fo rming r a t i o n a l expec ta t i ons e q u l i b r i u m . Chapter 3 Reve la t i on o f Aggregate Pre fe rence Parameters I n t roduc t i on In t h i s c h a p t e r , we study the ques t ion o f whether , i n a m u l t i -pe r i od economy, cu r ren t p r i c e s revea l enough i n fo rma t ion about the aggregate pre ference 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 tu re p r i c e s . Th is i s done i n a r a t i o n a l expec ta t i ons con tex t : the demands l ead ing to the format ion o f p r i c e s must be c o n s i s t e n t w i th the i n fo rma t ion 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 mo t i va t i on comes from the Arrow [1953] paper on the opt imal a l l o c a t i o n o f r i s k - b e a r i n g . Arrow f i r s t showed tha t i n an economy w i th C commodities and S s t a t e s o f the w o r l d , SC s t a t e - c o n t i n g e n t ("Arrow-Debreu") s e c u r i t i e s cor respond ing to each state-commodi ty p a i r w i l l ach ieve a Pareto e f f i c i e n t e q u i l i b r i u m . He then e s t a b l i s h e d tha t the same a l l o c a t i o n can be ob ta ined by S+C markets : S markets f o r s t a t e - c o n t i n g e n t weal th 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 example, by Dreze [1970, p. 144] and Nagatani [1975] , i s t ha t t h i s reduc t i on i n the number o f markets w i l l r e s u l t i n the same a l l o c a t i o n on ly i f agents know what commodity p r i c e s w i l l p r e v a i l i n each s t a t e o f n a t u r e , before i t i s revea led and hence before the market opens to generate these p r i c e s . I f cu r ren t p r i c e s can revea l p re fe rence parameters , they may revea l these s t a t e - c o n d i t i o n a l f u tu re p r i c e s , a l l o w i n g an opt imal a l l o c a t i o n w i th a reduced number o f markets i n the manner env i s i oned by Arrow. Another l i n e o f mo t i va t i on i s prov ided by Grossman [1976] , i n which agents come to a market w i th d i v e r s e i n fo rma t ion about a r i s k y a s s e t and the market c l e a r i n g process aggregates tha t i n fo rma t i on and conveys the aggregate parameter, va lue , v i a p r i c e s i g n a l s . In t h i s c h a p t e r , agents come to market w i th d i v e r s e p re fe rences and the market p r i c e conveys an aggregate pre ference va lue 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 impor tant i n p r e d i c t i n g next pe r iod p r i c e s . A t h i r d l i n e o f mo t i va t i on comes from the recen t l i t e r a t u r e on 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 [1979 ] , Dybvig and Polemarchakis [1979] and Green, Lau and Polemarchakis [1979] . The genera l ques t ion addressed i n t h i s l i t e r a t u r e i s : ' "G iven 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 oppo r tun i t y to observe 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 ( ' r e c o v e r e d ' ) ? " For example, Dybvig s t u d i e s the c l a s s o f a d d i t i v e 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 u t i l i t y f u n c t i o n i s recove rab le g iven two Engel curves ( imposs ib le ) or g iven th ree Engel curves ( g e n e r i c a l l y p o s s i b l e ) o r g iven demands a long two rays through the o r i g i n (always p o s s i b l e , p rov i d i ng the s lope o f the u t i l i t y f u n c t i o n i s bounded as consumption goes to 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 unc t i ons w i th more than two goods (or 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 o f von Neumann-Morgenstern u t i l i t y from demands f o r l o t t e r i e s over a s i n g l e r i s k y a s s e t and f i n d s tha t the main c o n d i t i o n f o r r e c o v e r a b i l i t y i s t ha t marginal u t i l i t y be bounded as consumption goes to z e r o . There are two important characteristics of these recoverability results,, in contrast to the present research: 1. U t i l i t y functions are recovered from demand functions, which are not observable from price data in a general equilibrium setting (unless multiple observations are available for econometric estimation). 2. They require an i n f i n i t e number of observations to specify whole Engel curves, for example, unless the Engel curves are previously known to have a s p e c i f i c functional form, in which case only enough observations to evaluate a l l parameters are required. In contrast, we use a general equilibrium setting and recover information by observing one r e a l i z a t i o n of the economy. This i s obtained at the expense, of course, of having to previously specify the functional form of the u t i l i t y function. The f i r s t models i n this chapter are i n a two period state prefer-ence setting 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 distributed security returns. Notation We deal with a pure exchange economy.and a single good ("money"). There are I agents ( i = 1, I ) , and three dates (0,1,2). Securities which are the only traded objects, pay o f f in money in C f i n a l (date 2) states (c = 1, ..., C) or in S intermediate (date 1) states (s = 1 There are homogeneous probability 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 states. These bel i e f s are: irs = Pr (intermediate state s) 7 T S C = Pr ( f i n a l state c|intermediate state s) i i E Pr ( f i n a l state c) = £ TT i r „ c s s sc One may think of the intermediate states s as presenting new proba-b i l i t y information causing each agent to update his probability estimates about f i n a l state c from T T c to i r s c . An agent i s endowed with money at date 0 and makes sequential p o r t f o l i o decisions at date 0 and date 1 f o r , respectively, an i n i t i a l p o r t f o l i o of claims on money in intermediate states and a f i n a l p o r t f o l i o of claims on money in f i n a l states. These are denoted by: y.j = endowed wealth of agent i y. • = payoff to agent i's i n i t i a l p o r t f o l i o i f inter-1 S mediate state s prevails at date 1 y. = payoff to agent i's f i n a l p o r t f o l i o i f intermediate state s prevails at 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 state c prevails at date 2. The aggregate supply of these claims i s X = Z|X 1- s e. Without loss of generality, we assume X i s independent of s. Markets consist of two types of s e c u r i t i e s : Type " f " is a r i s k l e s s bond maturing at 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 of the social endowment at the next date ( i . e . , the market p o r t f o l i o ) , with unit net supply. Define prices of the market p o r t f o l i o by: PQ = date 0 price of market p o r t f o l i o in terms of cash numeraire. Ps = date 1 price of market p o r t f o l i o in state s, in terms of a state s cash numeraire. Let m. = individual i's endowed fractional holding of the market p o r t f o l i o at date 0 (£.m. = 1), f i 0 = individual i's d o l l a r holding of cash in the p o r t f o l i o selected at date 0, m^ Q = individual i's fractional holding of next date social endowment in the p o r t f o l i o selected at date 0, f i s ' m i s E s i m i l a r l y for the f i n a l p o r t f o l i o selected at date 1 in intermediate state s. Then intermediate and f i n a l payoffs are, respectively, ( 3 J ) * i s = f i 0 + m i O p s <3-2> x i s c = f i s + m i s X c • Budget constraints for dates 0 and 1 are, respectively, (3.3) f i Q + p0miQ = p0m. (3.4) f i s + psmis = y. s . Market clearing relations are: (3.5) Z.f.Q - Z,fH - 0 (3.6) ^ i 0 = Z i m i s = 1 -There i s no consumption before the f i n a l period (and hence no natural inter-temporal discount rate), so that, in market M, there i s a need for S numeraires for date 1 and one numeraire for 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 for that date and state. 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 + p0miQ = y i . We may think of t h i s market setting as a r i s i n g at each date from complete markets of securities paying o f f at the next date. The complete market s t r u c t u r e degenerates to a market f o r two 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 t o l e rance (LRT) w i t h the same s l o p e . That i s , i f an agent has von Neumann-Morgenstern u t i l i t y U(x) f o r weal th x , then h i s r i s k t o l e r a n c e (at x) i s the i nve rse of h i s abso lu te r i s k a v e r s i o n : 1 = - U ' (x ) R A ( x ) " U"(x) We s h a l l assume tha t agents have l i n e a r r i s k t o l e rance w i th the same s lope c o e f f i c i e n t ( " c a u t i o u s n e s s " ) , i . e . : (3.8) U . ' ( x ) " U^T*) = a i + X x This i s necessary and s u f f i c i e n t f o r l i n e a r sha r i ng r u l e s and sur roga te f unc t i ons ( c f . Wi lson [1968]) and hence r e s u l t s i n aggregat ion ( c f . Rub ins te in [1974] and Brennan-Kraus [1978]) as we l l as two-fund monetary sepa ra t i on ( 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 w i l l degenerate i n t o t r a d i n g i n two p o r t f o l i o s : cash ("money") and the market p o r t f o l i o o f a l l r i s k y a s s e t s . We may assume, w i thout l o s s o f g e n e r a l i t y , tha t cash has a net zero s u p p l y , because the n e t . supp ly o f cash can be absorbed i n t o the market p o r t f o l i o , merely s h i f t i n g i t s r e tu rn d i s t r i b u t i o n by a cons tan t . A power u t i l i t y economy Suppose a l l agents have extended power u t i l i t y (wi th the same exponent) f o r consumption a t date 2 . S p e c i f i c a l l y , the u t i l i t y f o r agent i f o r consuming x a t date 2 i s : (3.9) U.(x) = Y " 1 ( e i + x ) Y i = 1 I where 0 f y < 1. This i s a concave, increasing u t i l i t y function displaying decreasing absolute r i s k a v e r s i o n . ^ At date 1 in state s agent i chooses a p o r t f o l i o (fn-s» to solve: max (e.+f. +m. X j Y ' C SC v 1 IS IS c < f i s ' m i s } subject to: (3.4) f i s + p s m i s =y. s . Note that ^ e have used (3.2) to substitute x i s c = f ^ s + m i sX c into the objective function. The f i r s t order conditions y i e l d h - i (3.10) p S c n s c X c 0 i+y i s+m . s (x c-p s)J e i + y i s + m i s ( X c - P S ) ¥-1 This i s a nonlinear equation i m p l i c i t l y giving no closed form solution r, i 2) . for the demand m s^ as a function of p $ and y ^ . Howeve  i t i s easy to ver i f y that (3.10) holds for a l l i i f and only i f (3.11) Q i + y i s i = 1, I and m-„ „ •, - • is 0 A + P s ( 3 ' 1 2 ) i n xfe.+xj^ - 1 P = c s c c A c where S Vsc ( 0 A + X c ) Y (3.13) 0 A = E I 0 i . Summing (3.1) over i and noting (3.5) and (3.6) yields ^ ^ y i s = P s Hence, summing (3.11) over i yields E . rn.s = 1, so that the market for the risky asset clears at date 1. By Walras' law, the market for the r i s k l e s s asset also clears, so (3.11) and (3.12) represents an equilibrium, where f . g i s computed from (3.4). Note that (3.11) does not g i ve 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 i t so l ves (3 .10) on ly when (3.12) h o l d s . However, i n e q u i l i b r i u m , (3.12) does ho ld and (3.11) g ives the c o r r e c t sha r ing r u l e or numerical va lue of demand. Another way o f o b t a i n i n g (3.12) i s to compute i t as the marginal r a te 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 asse t f o r a r e p r e s e n t a t i v e or aggregate i n v e s t o r A w i th a l l market weal th ^so tha t m A s = 1 and f A s = 0 by (3.5) and (3 .6 ) ) and wi th u t i 1 i ty U A (x) = Y _ 1 ( Q A + X ) Y where G A i s g iven by ( 3 . 1 3 ) . Hence, i t i s app rop r i a te to r e f e r to 0 A as a market r i s k t o l e rance parameter. At date 0 , a l l unce r t a i n t y about r e l a t i v e p r i c e s p g i n date 1 a r i s e s from e i t h e r unce r ta i n t y about the s t a t e s o r unce r t a i n t y about © A , 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 s t a t i s t i c " f o r 0-j, . . . , 0 j . We can now compute the date 1 de r i ved u t i l i t y f o r wea l th y ^ s , f o l d back to date 0 and so l ve f o r date 0 demands (and hence p r i c e s ) i n the manner o f dynamic programming. S u b s t i t u t i n g (3.4) and (3.11) 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 de r i ved u t i l i t y . ^ • E c n s c 0 A + X c l Y e A + p s (©•+y- ) As i n the d i s c u s s i o n o f Chapter 2 , suppose tha t 0 A and hence p g are known to a l l agen ts . We w i l l check to see t ha t i t s r e v e l a t i o n can be sus ta i ned i n e q u i l i b r i u m . Then the date 0 p o r t f o l i o problem becomes: max Y - 1 L n p ' s s { f . ,m. } 10 10 z c n s c V X c f e A + p s ( e i + y i s ) " y . = f . + m. p 7 i s i o I O ^ S f . + p m. = y . i o yo i o Ji sub jec t t o : (3.1) (3 .7) Th is i s an extended power u t i l i t y problem tha t i s e x a c t l y analogous to the date 1 problem when the p r o b a b i l i t y weights n s c are replaced";by n § t imes the f a c t o r . i n l a r g e square brackets 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 by : G ^ y . i = 1, . . . , I ( 3 ' 1 4 > m i o = e f l + P A p o Z 0 IL s s E c n s c [e A + X c ] Y [ V s J ; s s E c n s c [e A + X c ] r G A + P s j C a n c e l l i n g f a c t o r s i n ( e A + p s ) ? s u b s t i t u t i n g f o r p g from (3.12) and r e c a l l i n g tha t s n n = n y i e l d s : O O W W Vrf (3.15) p r V c X c WH  s c f c ( 0 A + X c ) Y _ 1 This y i e l d s p Q as a f u n c t i o n o f 0 .^ I f the f u n c t i o n i s i n v e r t i b l e then a f u l l y in fo rming r a t i o n a l expec t ions e q u i l i b r i u m (FRE) e x i s t s , as d i scussed i n Chapter 2. We have: Z C I I c ( e A + X c ) Y-1 2 dp W c ( 0A + Xc ) Y" 2 E d n d ( Q A + x d ) Y _ 1 '0 _ " £c,A«d ' 9 f l + X c ) Y " 2 ( e A + X d ) Y " 2 C Xc«V Xd> - < e A + X c » l = \ x c , d ¥ d <Vxc>Y"2 ( 0 A + x d ) Y " 2 C ( x c - x d ) ( x d - x c ) ] < 0 . Thus, pQ i s an in v e r t i b l e function of eft and we have . Theorem 3.1 If 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 exists a f u l l y informing rational expectations equilibrium (FRE) i n which agents can infer aggregate r i s k preference e^from the date,0_price pQ and hence correctly i n f e r the prices pg that would occur i f state s occurred (s = 1, ...., S). This model i s somewhat akin to the Grossman [1976] model in which individuals 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 distributed observations about the next period mean value of a risky asset's normally distributed return. Grossman showed that the market price i s an in 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 of his model that was pointed out by Grossman-Stiglitz [1976] and Kraus [1976] was that, since prices convey a l l relevant information, individuals w i l l not c o l l e c t private information i f i t i s costly to obtain. But i f they do not c o l l e c t private information, i t w i l l never be impounded in prices, so that prices w i l l not convey a l l available information after a l l . The model presented here does not have this d i f f i c u l t y , since agents must assess the i r own u t i l i t y functions before coming to market, in order to formulate the i r demands. This e f f e c t i v e l y costless, heterogeneous information w i l l come to market. ' Corollary 3.2: Under the same hypotheses as for Theorem 3.1, but assuming a l l agents have extended power u t i l i t y with increasing absolute r i s k aversion, v i z . (3.16) u\(x) = - ( e r x ) Y i = 1, I where y > 1, there exists a f u l l y informing rational expectations equilibrium where agents can in f e r from the date 0 price p Q . Proof: By reasoning analogous to that of Theorem 3.1, we have, for example, 0. -y • m i o - f j A - L ? X, ( 0 n - X j Y _ 1 p = c c c v A c' o V c ( e A - X c > Y _ 1 and ^ o > 0 5 ) . Q.E.O. d 0 A A natural question to ask i s how the results are affected by consumption at dates 0 and 1, as well as date 2. Suppose a l l agents have intertemporally additive power u t i l i t y functions, so that von Neumann-Morgenstern u t i l i t y becomes: Y " 1 ( V x i 0 ) Y + y" 1 1,^5115 ( e ^ ) * + Y ' 1 n 2 ^ c n s n s c ( G i + x i s c ) Y where agent i consumes x^ , x^ and x^ $ c respectively at 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 states, 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 in appropriate settings, yields 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 risky asset. (In fa c t , two assets w i l l do at each date, since the vector of agents' consumption good holdings in &1 w i l l be spanned by the vectors of I 6) r i s k l e s s asset holdings and risky asset holdings in £ . Date 0 re l a t i v e prices w i l l reveal 0^. However, the date 0 r i s k l e s s asset w i l l have a special form: i t must provide one.unit of r i s k l e s s consumption at date 1. Since the r e l a t i v e prices of these two types of consumption ( i . e . , the rate of interest) w i l l , in general, be diff e r e n t in di 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 variable payoff in 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 prices, which depend on 0^. Thus,;-not knowing 0 ,^ in complete markets an agent cannot compute which weights to use on prices, in order to compute the r e l a t i v e price of the risky and r i s k l e s s assets, and hence invert to get 0 .^ At date 0, there would be S prices of the financial securities 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 single parameter Q ^ , under many conditions, but a study of these does not appear to be very instru c t i v e . Intermediate labor income Another question i s how the results are affected i f date 1 (labor) income i s introduced for agent i . S p e c i f i c a l l y , suppose that, at date 1, agent i receives 1 income of $L., independent of state s, the value of which i s revealed at date 0 to agent i , but to nobody else. Thus, date 1 aggregate wealth i s uncertain, at date 0, creating more uncertainty about date 1 prices. We shall use the same notation as before with the following changes: The date 1-budget constraints become, ; . (3.17) f. + p m. = y. + L. v ' i s M s i s J i s i The date 1 market for the risk!ess asset clears when (3.18) E . f. = E.L. = L -| i s 1 1 The reason for having a nonzero aggregate supply of the r i s k -less asset i s to ensure that, by purchasing at date 0 the fraction mi>0 of the r i s k y asset, paying o f f p at date 1 in state s, agent i i s only making a claim to the risky market asset, rather than other agents' labor income. That i s , summing (3.17) over i and using (3.18) yields Ps = z.y i s rather than PG = z ^ y ^ + U which would be the case i f f.. = 0, which would be aest h e t i c a l l y , unappealing. In this s e t t i n g , the following holds: Theorem 3.3: Under the same assumptions as for Theorem 3.1 (ex-tended power u t i l i t y , e t c . ) , but substituting (3.17) and (3.18) for (3.4) and (3.5), respectively, where L. i s agent i's date 1 labor income, known to only him at date 0, there exists a f u l l y informing rational expectations equilibrium in which the date 0 price pQ reveals 0^  + L, which allows computation of date 1 prices p $ . Proof: Using the techniques of the proof of Theorem (3.1), one can show that: _ V s c X c < 6 A + L + X c ) Y " ]  P s ' E c n s ( G A + L + X c r ] and : 0- + L. + y. m ; = - J ! L i ns 0 A + L + p s At date 1 in state s , the u t i l i t y for f inanc ia l wealth y.. i s , f o r agent i , Y Y ( 0 i + L i + ^ i s ) (3 .19) z G n s c e A + L + X c •eA + L + p s I f agents know 0 A + L , the date 0 p r i c e i s ( e . + L + x J Y _ 1 ' c " c A c V U A ' u ' A c  p o " ^ c n c ( e A + L + X C )Y -1 and 0. + L. + y . i . I I i o 0 A + L + p Q Thus, a l l p r i c e s are a f u n c t i o n o f 0 A + L i n the same way tha 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^. J u s t as p Q r evea l s 0 A i n Theorem 3 . 1 , p Q r evea l s © A + L h e r e , and a FRE e x i s t s . Q.E.D 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 i s to suppose tha t L i i s revea led 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 ha t L = E^L^ i s not random, but known to a l l agents a t date 0 . Such a s i t u a t i o n cou ld a r i s e , f o r example, i f agents are s tevedores who repo r t to a h i r i n g h a l l every p e r i o d . Workers are ass igned to jobs randomly, s i n c e there i s not enough work f o r a l l . However, a l l workers know beforehand the t o t a l amount o f l abo 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 un ion . This minor change prevents 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 expec ta t i ons e q u i l i b r i u m tha t i s not f u l l y r e v e a l i n g . A FRE would have agents ho ld ing a t date 0 the f r a c t i o n : m. 9 -+L.+y. ' io e A + L + P o of the r i s k y a s s e t (from the proof o f Theorem 3 . 3 ) . I t would a l s o a l l ow them to compute + L from p Q , so i t would r equ i r e the random demand m. to generate the whole i n fo rma t ion a - a l g e b r a L..., a l though t h i s i n fo rma t ion i s not . a v a i l a b l e to any agent a t date 0. Hence a FRE cannot e x i s t . In t h i s market s t r u c t u r e , (3.19) s t i l l represen ts agent i ' s date 1, s t a t e s de r i ved u t i l i t y f o r weal th y i s , s i n c e i s known then . At date 0 , agent i must take expec ta -t i o n s over s t a t e s and , as we l l as over e ^ , p s , and y ^ s = m - j S + f - j 0 P s s i n c e a FRE d o e s n ' t e x i s t . The expected u t i l i t y becomes: JL£„iL I L „ E s c s sc fe A +L+ x J V L + p s (e.+L.+y. ) Y  v i i • ' i s ' e. ,p i 'Ho The expec ta t i on opera to r E i s the expec ta t i on c o n d i t i o n a l on p Q and e.-, w i t h r espec t to the random v a r i a b l e s e f l , p_, y ._ and , e . . Note tha t p Q p rov ides no i n fo rma t ion about L . , s i n c e no agents have in fo rma t ion about a t - d a t e . 0;'. Tf a r a t i o n a l expec ta t ions e q u i l i b r i u m e x i s t s , i t may be assumed tha t the 1 . are independent o f e f l , y . and 1 A I S p $ , 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 P q . The randomness o f serves to make markets i ncomp le te , thereby d e s t r o y i n g the sepa ra t i on and 43. aggregat ion p r o p e r t i e s . Another 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 " assumption about agen ts , analogous to the p r i c e - t a k i n g assumption used i n the theory o f compe t i t i ve markets . That i s , one can assume, as an approx imat ion from some law o f l a r g e numbers, t ha t the d i s t r i b u t i o n o f (e.+L+x )/(efl+L+p ) 1 C M S i s independent of tha t o f 6^ , even though 0 A = 6^. That i s , i f there are enough agen ts , agent i , w i l l not make any i n fe rences about p „ from h i s own e . so t ha t p n w i l l depend on e . on l y through 6„ Computing the f i r s t o rder c o n d i t i o n s and s u b s t i t u t i n g f o r from the budget c o n s t r a i n t y i e l d s (wi th endowment y^ = m^P0)> i n the general c a s e , rt.. „ > \y z z n n E i s c s sc 6 A + L + X c l 9 A + L + p s i P s ( 6 i + L. +m.p 0 +m i 0(p s-p 0)) Y-1 P 0 , e. ,m i s c s sc e A + L + X c 6 A + L + P S fe . + L . + m . p „ + n i . „ ( p ^ - p „ ) l Y-1 ( i = 1 , . . . , I ) . We a l s o have e n = i . e . and i , . m . = 1. Thus, there a re I+l c o n d i t i o n s M '• 1 1 1 1 0 on the endogenous v a r i a b l e s m^Q and p Q t ha t d e f i n e them, sub jec t to a c o n s t r a i n t on the exogenous v a r i a b l e s , and . By s u b s t i t u t i n g f o r from the budget c o n s t r a i n t , we deal w i th one market on ly so Wa l ras ' Law does not reduce the i n fo rma t ion 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 expec ta t ions 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 , but 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 po i n t prob lem, which may be s t u d i e d numer i ca l l y by computer. > FRE's w i t h o ther 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 A na tu ra l ques t ion i s whether a FRE e x i s t s when agents have u t i l i t y f unc t i ons i n the o ther c l a s s e s e x h i b i t i n g l i n e a r r i s k t o l e r a n c e , namely the extended l og and exponent ia l c l a s s e s . The answer i s a f f i r m a t i v e , as i n the next two theorems: Theorem 3.4 Suppose the market s t r u c t u r e o f Theorem 3.3 h o l d s , except t ha t agents a l l have extended log u t i l i t y f o r date 2 weal th o f the form U n.(x) = li) (e.j+x) Then a FRE e x i s t s i n which agents can i n f e r eA+L from p , and > hence can a l s o i n f e r the p r i c e s tha t w i l l ob ta in at date 1. P roo f : The f i r s t o rder c o n d i t i o n s are 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 : i n a d m i s s i b l e parameter va lue f o r power u t i l i t y ( ig iv ing cons tan t u t i l i t y ) . The proof o f Theorem 3.1 a p p l i e s w i th y = 0 , needing on ly elementary m o d i f i c a t i o n s . Q.E.© Theorem 3.5 Suppose the market s t r u c t u r e o f Theorems 3 .3 and 3.4 h o l d s , except tha t a l l agents have exponent ia l u t i l i t y o f the form U n.(x) = - exp(-G i X). . . -, -1 Then a FRE e x i s t s i n which agents can i n f e r o E ( Z ^ . ' ) from p Q and hence can a l s o i n f e r the p r i c e s tha t w i l l ob ta in a t date 1. Remark: A t date 0 , agents cannot and need not make any i n fe rences about aggregate date 1 l a b o r income L. E s s e n t i a l l y , t h i s a r i s e s because the aggregate i n v e s t o r has exponent ia l u t i l i t y and hence constant absolute r i s k aversion. Wealth does not affect choices among gambles, and hence does not affect the marginal rate of substitution between the risky and safe assets. Proof: At date 1 in state s, the f i r s t order conditions for agent i y i e l d : p s = E„IL„ .X. exp c sc c v c sc exp -0. (f. +m. X J 1 i s i s c' -0 . ( f . +m. X ) "I I S I S c = £c nSC XC e X P ( - 0 i m i S \ \ Vsc e x p ( L 0 i m i s V Note that the factor in f. i s j u s t a wealth effect and drops out is The f i r s t order conditions are the same for a l l agents when 0 i m i s = c o n s t a n t = 1 Since z. mis = 1, we have the sharing rule: - 9 m. = — is 0. and the price becomes £ c n s c X c e x P > 0 X c > z c n s c e x P (- 0 Xc^ After computing the derived u t i l i t y for date 1 wealth + y i $ , the date 0 f i r s t order condition becomes, for agent i , > . Z s E c n s n s c e x p ^ ^ s ^ c ^ p s , e x P K j ^ i o ^ i ^ i o P s ^ ° ~ V c n s n s c e x P l e < V V J e x p i-¥ fio + Li + m.io ps)) . Once again, the factors in f . Q and L i cancel and the market clears with the sharing rule: io ' i and p r i ce : V c V s c p s e x p ( - 6 X c }  P ° V c n s n s c e x P ( - e x c ) E H X_ exp (r& X j = c c c r c , - . E c n c e x p ( - e x G ) D i f f e r e n t i a t i n g p Q w . r . t e 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 on ly 0 and not L i s needed to compute p $ . Q.E.D R o l l o v e r o f p o r t f o l i o s a t date 1 A re -examina t ion of the proof 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. = m. ( i = l , . . . , I; s = 1 S ) . 10 1 TS Thus agent i w i l l t rade a t date 0 to a p o r t f o l i o c o n s i s t i n g o f the p ropo r t i on 0/0^ o f the r i s k y a s s e t and the remainder o f h i s weal th i n the sa fe a s s e t . A t date 1 i n s t a t e s , t h i s w i l l be worth P S (0/0.j) which i s j u s t enough to purchase the same f r a c t i o n o f 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 opt imal h o l d i n g . The date 0 r i s k ! e s s asse t ho ld ings f . Q are added to l a b o r income L. to g i ve the opt imal r i s k l e s s asse t ho ld ings f.. f o r date 1. Thus, a t date 0 , agent i may view the r i s k y asse t not as an a s s e t paying pg a t date 1 i n s t a t e s , but as an asse t paying X c a t date 2 in s t a t e ( s , c ) . . . Th is w i l l a l l ow him to t rade to an opt imal demand ho ld ing m ^ , which he merely needs to r e - i n v e s t mechanically ("rollover") at date 1 into the same amount of the ris 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 two stage problem. Using this p o r t f o l i o r o l l o v e r technique, agents need not predict date 1 prices p , and hence have no need to know e. The machinery used to establish the existence of a FRE i s not r e a l l y needed because agents can achieve optimal holdings using the rollover technique. Interpreted broadly, this r o llover economy offers a FRE in 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 different u t i l i t y function w i l l not choose to roll o v e r holdings, but w i l l desire to trade at date 1. For such an investor, i t i s important to in f e r the aggregate risk aversion that sets prices. The question arises as to whether or not the same r o l l o v e r technique works for extended power and log u t i l i t y economies. For these economies, the optimal risky asset holdings at date 1 (state s) and date 0 are, respectively: m i s V ^ i s eA+L + ps and, m.o 9 i + L i + y j Q A + L + po Expanding m^s» we find that: — — l - — ! — L y _ £ _ ^ by (3.7) eA + L + ps m. (e.+L+p +P -P ) 1 0 H " b u s u b s t i t u t i n g m. from eA+L+P 1 0  M s above. 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 present i n the power and l og u t i l i t y economies, as w e l l . The r o l l o v e r a l go r i t hm a r i s e s because ho ld ings i n the market p o r t f o l i o a t date 0 can be r o l l e d over i n t o i d e n t i c a l ho ld ings i n any s t a t e a t date 1. The same holds f o r the r i s k l e s s a s s e t : $1 a t date 0 b r ings $1 i n whatever s t a t e s occurs a t date 1, which b r ings $1 i n whatever s t a t e c occurs a t date 2 . (With l a b o r income, the agent merely counts the date 1 as pa r t o f "human wea l t h " at date 0.) Hence, a t date 0 , agents can e f f e c t i v e l y buy c la ims 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 . S ince these are the on ly s e c u r i t i e s they would purchase i f presented w i th a complete s e t o f date 2 con t ingen t c l a i m s , they e f f e c t i v e l y face a complete market i n date 2 con t ingen t c l a i m s , a l though a t date 1 they r e v i s e t h e i r es t imates o f the p r o b a b i l i t i e s o f the occurrence o f the f i n a l s t a t e s . H i r s h l e i f e r [1971] and Marsha l l [1974] have shown tha t the c o n t r a c t curve o f an exchange economy w i th 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 tha t a t date 0 , agents t rade 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 fo rma t ion a t date 1. Note tha t t h i s r o l l o v e r f ea tu re 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 rance u t i l i t y f u n c t i o n s . 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 i s i n a general e q u i l i b r i u m s e t t i n g , wh 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 curve and merely r o l l i n g over t h e i r p o r t f o l i o s 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 investment 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 . , i f 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 d i s t r i b u t i o n s must change, no mat ter how p r i c e s move). Th is p rov ides a coun te r -example to M o s s i n ' s [1968, p.122] con ten t ion tha t agents w i l l have s t a t i o n a r y investment p o l i c i e s on ly i f y i e l d d i s t r i b u t i o n s are s t a t i o n a r y . t h i s r o l l o v e r f ea tu re cannot be avoided i n a s t a t e p re fe rence s e t t i n g w i thou 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 osed form p r i c e s a r i s e from the aggregat ion and sepa ra t i on i n the 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 the e f f e c t i v e l y complete markets a t dates 0 and 1. Without complete markets , aggregat ion f a i l s , but w i th complete marke ts , 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 o b t a i n s . Th is mot iva tes the model o f the next s e c t i o n i n which date 0 markets are no t comple te , but sepa ra t i on ob ta ins because re tu rns are normal ly 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 us ing cons tan t abso lu te r i s k ave rs i on (exponen t ia l ) u t i l i t y . 50. A model i n which date 0 p r i c e s revea l aggregate pre ferences and agents r e -ba lance p o r t f o l i o s i n the in te rmed ia te pe r iod As i n the prev ious s e c t i o n s , cons ide r a pure exchange economy w i th I agents and three dates (0 , 1, 2 ) . A t date 0 there are two s e c u r i t i e s which have j o i n t l y normal ly d i s t r i b u t e d date 1 payof fs r r w 1 w. ~ N w 1 a\ 0 0 al • f \ a i l 1 • a i 2 v. J 1 At date 0 agent i s e l e c t s a p o r t f o l i o vec to r ( c t ^ - j , a - j 2 ) ' a n c * r e a l i z e s re tu rns a t date 1 o f = a.^W-j + a i 2 W 2 . He faces the budget c o n s t r a i n t (3 .20) y i Q = a „ + P 0 a i 2 where y i Q i s h i s i n i t i a l endowment and p Q i s the p r i c e o f the second asse t (w i th the f i r s t as numera i re ) . Markets a t date 0 c l e a r when (3.21) - l At date 1 , there i s a r i s k l e s s asse t w i th re tu rn R and a r i s k y a s s e t w i th a date 2 payof f o f V ~ N(V, Oy) Agent i r e a l i z e s date 2 weal th o f y i 2 S 3 i R R + 3 i v V where e i R i s h i s l end ing /bor row ing and e i v i s h i s ho ld ing o f the r i s k y a s s e t . He faces the budget c o n s t r a i n t (3.22) y i l = 6 i R + P ^ i v where p-j i s the p r i c e o f the (second per iod ) r i s k y a s s e t (wi th the r i s k l e s s a s s e t as numera i re ) . The market c l e a r s when (3.23) I • 0 "1V \ J 1 » 4 I n d i v i d u a l i has u t i l i t y f o r f i n a l weal th o f • - e x p ( - e . y i 2 ) • A t d a t e 1 he maximizes e i 2 2 - E exp ( -e i ( /3 . R R^. v V) : . ) =• -exp(-0 i(/3 i RR+/3. vV)+ — ^. v a v ) sub jec t to ( 3 . 2 2 ) . S o l v i n g (3.22) f o r , 0 . R and s i m p l i f y i n g the ob -j e c t i v e y i e l d s the f i r s t o rder c o n d i t i o n e 2 e ^ V - p ^ - e ^ a 2 ) . e x p ( - " e . ( 0 i R & t f 1 v V ) + ~$\^\) = 0 . Hence (3.24) ^iv - < V-P{R Summing over i and no t ing (3.23) y i e l d s ( 3 - 2 5 ) p, - •*> where 0 = (£ 0 ~^)~^ . Th is g i ves the c a p i t a l a s s e t p r i c i n g model i i (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 ) , the de r i ved u t i l i t y f o r date 1 weal th y ^ l becomes - e x p t - G ^ y ^ ) exp(-ia9 20 2) . 2 Assume t ha t R i s known and f i x e d beforehand, and tha t o-y i s nons tochas t i c so t ha 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 ha t may be i gno red . Thus, a t date 0 , i n d i v i d u a l i maximizes 0 2 R 2 - - i 2 2 2 2 - E e x p f - e ^ R y ^ ) = -exp(- 'e .R(a i 1 i W.| + a I 2 W 2 ) + — — ( a ^ ^ + a ^ a 2 ) ) s u b j e c t to the budget ( 3 . 2 0 ) . S u b s t i t u t i n g f o r from the budget , the f i r s t o rder 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 " h + 0 i R ( - p o ( y i o ' P o a i 2 } a l 2 + a i 2 a 2 2 ) = 0 Div i de by and sum over i , no t ing from (3.20) and (3.21) tha t Vio = 1 + p o t 0 g e t W IP Q - W 2 + 9 R (-P 0 ( l +P 0 ) o - , 2 + P Q O - , 2 + a 2 2 ) = 0 V o " W 2 so tha t e = - 1 0 L R ( o 2 2 - P 0 C 2 ) 7 7 Since , W 2 , a-j , a 2 and R are known, one can compute e knowing p Q , so t ha t a F R E e x i s t s and s i n c e date 0 markets are i ncomp le te , agents must r e -ba lance t h e i r p o r t f o l i o s a t date 1. Th is y i e l d s the f o l l o w i n g Theorem 3.6 In a two pe r iod ( three date) economy, where the second pe r i od market c o n s i s t s o f a normal ly d i s t r i b u t e d r i s k y asse t and a r i s k l e s s asse t (wi th known i n t e r e s t r a t e ) , the f i r s t pe r iod market c o n s i s t s o f two normal ly 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 exponent ia l u t i l i t y , as desc r ibed 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 ave rs i on parameter 0 which i s revea led by date 0 p r i c e s . Hence a f u l l y in fo rming r a t i o n a l expec ta t i ons (FRE) e x i s t s , even though date 0 markets are incomplete and agents must r e -ba lance t h e i r p o r t f o l i o s a t date 1. The t r a c t a b l e computat ions i n Theorem 3.6 r e s u l t e d p a r t l y from the p o r t f o l i o sepa ra t i on tha t was induced by the normal ly d i s t r i b u t e d p o r t f o l i o r e t u r n s . Date 0 markets do not have a r i s k -l e s s a s s e t and hence are incomplete so tha t the exponent ia l u t i l i t y f u n c t i o n i t s e l f i s not enough to ensure p o r t f o l i o s e p a r a t i o n . Th is suggests t ha t the theorem may a l s o extend to o ther u t i l i t y f unc t i ons f o r which the u t i l i t y o f negat ive weal th i s w e l l - d e f i n e d . (The normal d i s t r i b u t i o n has suppor t on the whole r ea l l i n e . ) Th is e l i m i n a t e s u t i l i t y f unc t i ons such as extended l og and power w i th i r r a -t i o n a l or the ' i nve rse of even exponents. For-power u t i l i t y o f the form U!j(x) = Y ^ ( Q i + X ) Y W I T N Y = m / ( 2 n + 1 ) ; m,n i n t e g e r s , the ex-pected 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 of two gamma f u n c t i o n s . Th is separa tes but does not aggregate i n the date 0 market , and does aggregate i n the date 1 market ^ H e n c e , date 1 p r i c e s depend on = z-j0-j> but the date 0 p r i c e depends on a d i f f e r e n t f u n c t i o n o f (e-j, 0j) . In t h i s c a s e , i t i s doubt fu l whether a FRE e x i s t s , so tha t Theorem 3.6 may not be robust w i th r espec t to r e l a x a t i o n o f the exponen t ia l u t i l i t y c l a s s assumpt ion. Th is l i n e o f reasoning a l s o suggests tha t i t i s not adequate 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 tha t ensure p o r t f o l i o s e p a r a t i o n , as i n Ross [1978b] , to ensure t r a c . t a b i l i t y , . s i n c e the aggregat ion r e s u l t s must a l s o o b t a i n to ach ieve the parsimony o f parameters . The mo t i va t i on 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 th and w i thou t a r i s k i ess a s s e t (and hence, w i th and w i thou t complete markets) to get two fund s e p a r a t i o n . The two fund sepa ra t i on r e -s u l t s w i thou t a r i s k l e s s a s s e t f o l l o w from B lack [1972] . For the CAPM, the market p r i c e o f r i s k can be expressed i n terms o f each agen t ' s " g l oba l r i s k a v e r s i o n " E UV ' t x ) E U . ' ( x ) ( c f . Rub ins te in [1973] ) . S ince t h i s i s cons tan t on ly f o r exponent ia l u t i l i t y , exponent ia l u t i l i t y was chosen f o r the model . I t would appear , t hen , tha t the on ly r ea l g e n e r a l i z a t i o n tha t t h i s model admits ( t ha t r e t a i n s the pars imonious aggregat ion and sepa ra t i on p r o p e r t i e s ) i s the g e n e r a l i z a t i o n to seve ra l m u l t i v a r i a t e normal a s s e t s . The r e s u l t s are s t r a i g h t f o r w a r d , s i n c e sepa ra t i on ob ta ins even w i thout the r i s k l e s s a s s e t (as i n the B lack [1972] CAPM). I t i s even hard to i nco rpo ra te incomplete markets i n the second pe r iod as we l l as the f i r s t , s i n c e then the de r i ved u t i l i t y f o r date 1 weal th has a q u a d r a t i c f u n c t i o n i n the exponent, y i e l d i n g a b l i s s po i n t o f maximum g loba l u t i l i t y . Such an economy would not be mean-i n g f u l w i thou t 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 de r i ved u t i l i t y becomes i n t r a c t a b l e . Conc lus ion The models o f t h i s chapter were formula ted to study whether c u r r e n t p r i c e s can revea l enough in fo rma t ion about p re fe rence parameters to r e s o l v e some o f the unce r t a i n t y about p r i c e s i n the next p e r i o d . For s i m p l i c i t y , a l l the models were cons t ruc ted so tha t on l y an aggregate p re fe rence parameter had to be conveyed. Chapter 4 dea ls w i th the o ther major exogenous f a c t o r (bes ides p re fe rences) t ha 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 , aggregat ion ob ta ins 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 tud ied i n a non-aggregat ion s e t t i n g . Footnotes to Chapter 3 . 1. One may regard as a r i s k t o l e r a n c e parameter , s i n c e (3.9) has dec reas ing abso lu te r i s k a v e r s i o n , so tha t i n c r e a s i n g 8. i s e q u i v a l e n t to i n c r e a s i n g weal th and hence decreas ing abso lu te r i s k a v e r s i o n , or i n c r e a s i n g r i s k t o l e r a n c e . 2 . I t i s va luab le to e s t a b l i s h t ha t there i s a unique p r i c e system tha t e q u i l i b r a t e s the economy, so- tha t (3.12) does rep resen t the e q u i l i b r i u m p r i c e system. I f p r i c e s are not un ique , agents must have some way o f knowing tha t e q u i l i b r i u m p r i c e s are represented by (3.12) r a t h e r than some o ther p r i c e system. In economies where a l l agents have u t i l i t i e s from one c l a s s tha t aggregates ( tha t i s , f o r which aggregate excess demand f unc t i ons are una f fec ted by r e d i s t r i b u t i o n s 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 are formed as though a l l wea l th were endowed upon a s i n g l e agent (wi th s u i t a b l e aggregate p r e f e r e n c e s ) , and, i n e f f e c t become "one household economies. " I f i n a d d i t i o n , demands are s i n g l e - v a l u e d , s u i t a b l y cont inuous and bounded from below, a one-household economy e q u i l i b r a t e s w i th a unique p r i c e vec to r (cf_. Arrow-Hahn [1971, pp. 217-220 ] ) . In f i n a n c e , p re fe rences aggregate i f f u t i l i t i e s are a l l e x p o n e n t i a l , o r a l l extended power w i th the same exponent or a l l extended l o g , as used i n Chapters 3 and 5 ( cf_. Rub ins te i n [1974] and Brennan-Kraus [1978] ) . These u t i l i t i e s y i e l d s u f f i c i e n t l y we l l -behaved excess demands so tha t p r i c e s are unique i n complete markets or markets where there i s p o r t f o l i o sepa ra t i on y i e l d i n g e f f e c t i v e l y complete markets (and hence admi t t i ng a g g r e g a t i o n ) . Another way to cons ide r t h i s , suggested by Stephen Ross , i s to s p l i t a l l weal th between two i d e n t i c a l 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 market , the c o n t r a c t curve i s , by symmetry, a l i n e . Th 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 pre ferences tha t lead to aggregat ion i n f i nance are a l l homothet ic ( through some p o i n t , perhaps - » ) , the i n d i f f e r e n c e curves a l l cu t the E n g l e - l i n e at the same angle so tha t on l y one p r i c e hyperplane can e q u i l i b r a t e the economy f o r any one se t of :endowments and p re fe rences . Agent 1 Agent 2 F igure 3 . 1 . Edgeworth box w i th l i n e a r c o n t r a c t c u r v e , homothet ic i n d i f f e r e n c e curves and endowment e . Checking t o - see whether the bases of the exponents used throughout t h i s a n a l y s i s are p o s i t i v e , note tha t we must assume 8 .+X_£0, c = 1 , . . . , C . By (3.12) and ( 3 . 1 5 ) , p >0 (s = 1 , . . . , S ) and p > 0 . A l s o , from (3.12) and ( 3 . 1 5 ) , z n (e.+x ) Y e.+ P c = , c s c A c -j > 0 E n (e f l+X ) Y " ] c sc A c z j i ( 0 f l + X r ) Y and e f l + P o = c c H c > ^ ( e . + X J ^ - 1 c c A c 0 A l s o , i n e q u i l i b r i u m , i.+ x . = e . + f . ' „+ m. X. i i s c i i s i s c ( 0 A + X ) 7 = (o.+ry. ) ,/X .... • • • i rs' (e A +p s ) Th is i s p o s i t i v e i f f 0 . + y > 0 , By s i m i l a r r e a s o n i n g , 0 . + y . = ( 6 . + y . ) w * 1 J i s v T . " V ( E A + P Q ) which i s p o s i t i v e i f f e .+ y . = 0 . + m.p > 0. S ince p > 0 , K i J-\ i i o 0 t h i s i s p o s i t i v e under a wide range o f c i rcumstances — e . g .  e i 1 0> 171 -j > 0- Note tha t one r a t i o n a l e f o r extended power and l og u t i l i t y i s tha t - 0 . > 0 i s a minimum subs i s tence l e v e l o f consumpt ion, so "that x ^ s c i s consumption beyond the subs i s tence l e v e l . Th is r a t i o n a l e f a i l s i f e.. > 0 . F i n a l l y , we note that for certain parameter values, some agents may go bankrupt i n some states ~ i . e . , have x. s 0 or y. < 0 for some i , s, c. As long as agents are f S 0 IS not allowed to default at date 2 i f x i s c < 0 (e.g., negative consumption of financial wealth i s feasible i f agents have human wealth that they can use to pay off debts at date 2), then, even i f y i s < 0 at date 1, they w i l l choose to hold a po r t f o l i o as indicated by the sharing rules (and an " i n t e r i o r " optimum for the i r f i r s t order conditions) rather than default at date 1, so long as e^y... > 0, as indicated in Figure.-3.2 below. (-e.,0) y i s<0 (0,0) f i s Figure 3.2 Interior optimum The indifference curves for f. and m. induced by the s t r i c t l y I J I •) concave von Neumann-Morgenstern u t i l i t y cross the mis 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 total f i n a l wealth. See also Theorem 3.3. I am endebted to S. Grossman for reminding me of this nice feature of the model. Now, we require en>X ,c = 1,...,C, for p and p to be n A c ; o s r e a l . As in footnote 3, th i s ensures that the bases of exponents w i l l be positive, as long as e.>y.,i = 1,...,I. For example, i f asset markets are complete at date 1 in state s, agent i can be viewed as facing r i s k y returns with S+l; states (consumption and the s previous risky states). There i s no problem resulting from the f a i l u r e of the pr 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 and i s i s i s s x. = f. + m. X , for some optimal f-c,m. . isc i s i s c i s i s where X g i s the aggregate supply of the consumption good in state s. Hence, only two vectors,' (f - | S , . . .f I s ) E ^ and ( m ^ s . .m^e'S1. are needed to span the space of a l l agents optimal holdings of risky and consumption assets. That i s , the date 1 market i s complete and yields aggregation and separation because of 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 distributed random variables at date 0 induce a mean variance model which, of course separates, even without a r i s k l e s s asset. 61. Chapter 4 Reve la t i on o f I nd i v i dua l Endowments  I n t r oduc t i on In the l a s t chapter 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 . p re ference data ( a c t u a l l y an.aggregate pre ference datum), which cou ld be used by agents to make i n fe rences about f u tu re p r i c e s . Here , we study whether p r i c e s can s i g n a l i n fo rma t ion about the d i s t r i b u t i o n o f endowments o f w e a l t h . S ince t h i s i n fo rma t ion i s n e c e s s a r i l y m u l t i v a r i a t e , we study c o n d i t i o n s under which a vec to r o f endowments can be s i g n a l l e d by p r i c e s . In o rder to i n f e r the va lue o f a vec to r i n Euc l i dean n-space ( £ a ) , one must, i n g e n e r a l , be ab le to observe a r e l a t e d vec to r i n &m where m ^ n. o For example, i f ( x , y , z ) denotes a po in t i n £ , one can i n f e r the p o s i t i o n o f the po in t us ing three s p h e r i c a l coo rd ina tes (two angles and a r ad ius ) or three c y l i n d r i c a l coo rd ina tes (an a n g l e , a he igh t and a r a d i u s ) , but never w i th two coord ina tes o n l y . In the absence of degenerac ies , i t i s necessary to have m >_ n i n o rde r to convey n-d imensional i n fo rma t ion w i th an m-vector . 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 tud ied 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 tha t c a s e , a FRE e x i s t e d . 0 M Most o f the r a t i o n a l expec t ions l i t e r a t u r e to t h i s date on l y d i scusses the e x i s t e n c e o f a FRE when n <_ 1. For 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 fo rma t ion (n=0) by p r i c e s , and Grossman [1976] has n=l (an aggregate i n fo rma t ion parameter ) . To e s t a b l i s h the ex i s t ence o f a FRE, one must show tha t a map from the parameter space (a subset o f & n ) to the p r i c e space (a subset o f a " ) i s i n v e r t i b l e . ( I f the s o c i a l endowment i s nonrandom, we drop one dimension from the p r i c e space , by Wa l ras / ' law. ) I t i s g e n e r a l l y much harder to show c o n s t r u c t i v e l y t ha t a map from £ n to £ m i s one- to-one when m - v n > 1 than i t i s when m = n = 1. The ana lyses o f m u l t i v a r i a t e FRE's are by A l l e n [1978, 1979] . She 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 proper ty (o f an economy) i s g e n e r i c i f a r b i t r a r i l y smal l adjustments to the parameters ( t a s t e s , endowments, p r o b a b i l i t i e s , e t c . ) do not des t roy the p r o p e r t y , and , g iven an economy w i thou t the p r o p e r t y , there are a r b i t r a r i l y c l o s e economies tha t e x h i b i t t ha t p rope r t y . Thus, a gener i c p roper ty i s open and dense i n the f am i l y o f economies under some app rop r i a te t o p o l o g y . ^ The A l l e n r e s u l t s r e l y on the Whitney embedding theorem o f d i f f e r e n t i a l topo logy ( see , e . g . , H i r sh [1976]) to get the g e n e r i c i t y o f the e x i s t e n c e o f a FRE when m >_ 2n + 1. Using a d i f f u s e n e s s assumption on p re fe rences 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 to m > n + 1, i f agents are not concerned tha t the p r i c e map i s many to one on a se t o f p r o b a b i l i t y z e r o . The r e s u l t s are not robus t w i th respec t to r e l a x a t i o n o f the requirement t ha t m >_ n + 1, f o r Jordan and Radner [1977] p rov ide a gener i c nonex is tence r e s u l t when m = n = 1. The use of d i f f e r e n t i a l topology i s somewhat noncons t ruc t i ve and r e s u l t s us ing i t are hard to grasp i n t u i t i v e l y . In o rder 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 l a r g e l y r e l y on the b e t t e r known i m p l i c i t f u n c t i o n r e s u l t s which are based on the rank o f the Jacob ian ma 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 on l y o f a l o c a l n a t u r e , so we s h a l l s tudy the e x i s t e n c e o f 63. l o c a l l y f u l l y informing rational expectations e q u i l i b r i a (LFRE) for which there exists some open set of information parameters such that economies r e s t r i c t e d to this 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 analysis i s based on a power u t i l i t y economy (where agents have dif 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 related p a r a l l e l economies to get the generic r e s u l t s . Market structure, notation and preferences As in Chapter 3, there are three dates, with S states at date 1 and C states at date 2. Transition p r o b a b i l i t i e s are, at date 0, n s (s = 1, ..., S), and at date 1, state 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^ g. At date 1, agent i trades to date 2 state c contingent wealth x.jSC> which he consumes, achieving u t i l i t y Exogeneous social endowments of wealth at dates 0, 1, and 2 are, respectively (4.1) V x i s c ) = Y i -1 x i s c Y i (Ofri<l) or U ^ x ^ ) = log x i s c (y-0) Y = (4.2) is (4.3) E_. X. I S C Note t h a t , h e r e , Y and Y g are both g iven exogenous ly , i n c o n t r a s t to the models o f Chapter 3 , where Y = p Q and Y g = p g were endogenous p r i c e s . The va lues o f the Y g are exogenously revea led to a l l agents a t date 0 , a l though agents do not know the va lue o f Y . Markets prov ide a complete s e t o f f i n a n c i a l c la ims to weal th con t ingen t on the s t a t e o f nature a t the next da te . P r i c e s a r e : q s = date 0 p r i c e o f a c l a i m on $1 a t date 1, con t ingen t on in te rmed ia te s t a t e s . P s c = date 1 p r i c e when in te rmed ia te s t a t e s occurs o f a c l a i m on $1 a t date 2 , con t i ngen t on f i n a l on f i n a l s t a t e c . I f agent i knows the p r i c e s ( p ^ > •••> p$rj> 9i> • • •» a t date 0 , he can so l ve the 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 sub jec t to max z c n s c U ^ s c ) { x i s c } <"4-4> E c Psc x i s c = y i s A t date 0 , he performs m a x V c n s n s c u i ^ x l s c ) { y i s > sub jec t to x i s c ° P t i m a l i n date 1, s t a t e s and (4 .5) E s q s y i s = y. Denote t h i s market regime by M. I t i s represented by the t ree i n F igure 4 . 1 . (date 0 market) sc (date 1 market) Date Figure 4.3 Market Regime M" I t i s also useful to consider two related market regimes. F i r s t consider the complete market M' for date 0 claims to money in date 2 state c, and for which there are no intermediate states. Let r £ E date 0 price of claim to $1 at date 2, contingent on f i n a l state c. X - j c = payoff to individual i's p o r t f o l i o i f f i n a l state c prevails at date 2 TT = - i n n , the. unconditional probability of state c occurring, The budget constraint of agent i i s (4.6) £ r x. = y. C C IC J l Demands by individual i are determined by (4.6) and the f i r s t - o r d e r conditions ( 4 . 7 , . ^ . u ' ld< x 1d ' " V ' d Equilibrium requires that demands s a t i s f y the market clearing condition (4.8) Z i X i c = X c Now consider a more refined complete market M" for 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 ide n t i t y of the intermediate state as well as the f i n a l state. Let p s c = date 0 price of claim to $1 at date 2, contingent on intermediate state s at date 1 and f i n a l state c at date 2. Agent 1!s- budget constraint i s 67. The f i r s t order c o n d i t i o n s f o r agent i a re U , ic ( x1sc> P s c / V s c u ' i d ( x i t d ) P t d / V t d ( s , t = 1 , . . . , S ; c , d = 1 , . . . , C ) The market c l e a r i n g c o n d i t i o n i s ( 4 . 3 ) . Observe tha t the e q u i l i b r i u m tha 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 i s c = x i c ( 4 - 9 ) P s c = ( V s c / \ ) r c • Markets M 1 and M" present the same oppo r tun i t y s e t to a l l i n d i v i d u a l s and so market va lues must be the same i n bo th . Assume M" has a unique 2) e q u i l i b r i u m . ' The reason f o r i n t r o d u c i n g markets M1 and M" as a s tep i n the a n a l y s i s o f the market M i s t ha t Arrow [1953] has shown t h a t , i f a l l i n d i v i d u a l s are informed o f ex post e q u i l i b r i u m p r i c e s i n market M, i n d i v i d u a l s w i l l f ace the same oppo r tun i t y se ts and ach ieve the same f i n a l s t a t e payof fs i n M and i n M" . S i nce M' and M" a re e q u i v a l e n t , we can ana lyze M through c o n s i d e r a t i o n o f the s imp le r , market M ' . Arrow has a l s o shown tha t 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 ^ s p s c ~ p s c * Summing i n d i v i d u a l f i n a l pe r iod budget c o n s t r a i n t s i n M and app ly ing the market c l e a r i n g r e l a t i o n s (4 .2 ) and (4.3) y i e l d s E „ p c J L = Y c c sc c s 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',0) % - K/'i^c Kc'V'A (4.11) P s c = ( V s c / ; c ) ( r c / q s > • The r e v e l a t i o n o f i n fo rmat ion Assume tha t a l l agents know tha t everyone has power or l og u t i l i t y o f the form ( 4 . 1 ) , and they a l l know tha t a l l agents have u t i l i t y exponents from the s e t Yj>- However, agent i does not know how much weal th y . the o the r agents ( j= l , . , . ' j f i ) have. There may be a c l a s s o f seve ra l agents w i th 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 be the same as i f a l l wea l th o f the c l a s s were bestowed on a s i n g l e agent w i th 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 though an agent knows h i s personal w e a l t h , i n g e n e r a l , he w i l l not know the aggregate weal th y . o f h i s c l a s s , so tha t a l l agents I 3) d e s i r e to i n f e r the whole vec to r o f weal ths (y- j , . . . , y j ) e £ . / At date 0 , agents do not know the y^ or the aggregate Y , but do know the aggregate s o c i a l endowments o f date 1 weal th {Y s> and date 2 weal th {X c> ., As i n Chapter 3 , they have homogeneous p r o b a b i l i t y b e l i e f s { ^ s ^ s c ^ * T n e y d e s i r e to i n f e r the va lue o f ( y . ) 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 s c ) -I t i s conven ient to cons ide r f i r s t the market M' i n t h i s s e t t i n g and then get q s and p s c i n M by us ing (4.10) and ( 4 . 1 1 ) . Agent i ' s demand f u n c t i o n a t date 0 in M' can be de r i ved as f 6 i l I V 1 - 6 i 1 E d ( r d / l T d } - i x i c = t J where -= ( y r i ) D i f f e r e n t i a t i n g t o t a l l y w i t h respec t to x . , y i and rQ, summing over i n d i v i d u a l s and no t ing t ha t X £ i s f i x e d , so tha t E-jdx.^ = dX G = 0 , 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 mat r i x form: (A - X'GBX) d r + X 'G dy = 0 where A = diag [ l ^ V " ^ X S [ x 1 c ] B = d iag [^+1] G = d iag [y. _ 1] dr = ( d r . , , . . . ,dr(0,' dy = (dy-, , . v . , d y I ) ' and " d i a g " denotes a d iagonal m a t r i x . Assuming p r i c e s are d i f f e r e n t i a t e f u n c t i o n s o f weal th endowments, (4.12) dr = - (A - X ' G B X ) " 1 X 'G dy . A l s o , (4.10) can be expressed as (4.13) q = EnF r where q = ( q 1 , . . : . , q ^ ) 1 E = d iag [TT S /Y S ] n = [ i r s c ] F = d iag [ X ^ ] r = ( r ^ , . . . , ) Hence, changes i n observed p r i c e s are r e l a t e d to changes i n weal th endowments by (4.14) dq = Hdy where dq = (dq 1 d q s ) ' H = -EnF(A-X , GBX)~ 1 X , G App ly ing the i m p l i c i t 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 o f ( q - | , . . . , q s ) i f the c o e f f i c i e n t mat r i x H has rank I. I f t h i s i s the c a s e , then p s c i s l o c a l l y a f u n c t i o n o f (q-| q^) because p i s a f u n c t i o n o f r and q by (4.11) and r i s a f u n c t i o n o f (y- ]> ' ">y j ) by the uniqueness o f p r i c e s i n economy M ' . Hence, a LFRE would e x i s t . Observe tha t EnF i s SxC, ( A - X ' G B X ) - 1 i s CXC, and X 'G i s C x i . C l e a r l y , rank (H) < I i f C < I o r S < I, so t ha t p r i c e s q cannot communicate a l l o f the I -d imens iona l -(y.) i n f o r m a t i o n . However, t h i s need not r u l e out the e x i s t e n c e o f a FRE. Moreover, even i f C >_ I and S _> I, i t i s s u r p r i s i n g l y hard to e s t a b l i s h c o n d i t i o n s f o r rank (H) = I. The problem requ i r es an a n a l y s i s by c a s e s . Case 1: 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 the e x i s t e n c e o f a FRE ( g l o b a l , not j u s t l o c a l ) i s t ha t rank (n) = C. P roo f : S ince E and F are n o n s i n g u l a r d iagonal m a t r i c e s , rank (EnF) rank (n) = C. Thus the mapping (4.13) o f r to q i s i n v e r t i b l e . By observ ing q , an agent may i n f e r r and hence { p c r } by ( 4 . 1 1 ) . Th is a l l ows him to s e l e c t h i s date 0 p o r t f o l i o i n M as though he were f u l l y in fo rmed. Q . E . D . Note tha t i n Theorem 4.1 agents do not l e a r n a l l i n fo rma t i on about y = ( y - j . . , y j ) , but j u s t enough about i t ( i . e . , about which r and { p c r } ob ta i ns ) to behave o p t i m a l l y . Case 2 : S < min ( I , C ) . With I > S and C > S there are not enough dimensions o f v a r i a t i o n i n (q-, q^) to in fo rm market p a r t i c i p a n t s o f the va lue o f e i t h e r ( r . , , . . ' . . r^ ) o r (y- | , . • • , y s ) . Thus, we have: Theorem 4 . 2 : I f S < min ( I ,C ) n e i t h e r a LFRE nor FRE e x i s t , un less the market i s degenerate to a l l ow aggregat ion o f i n d i v i d u a l s ( thus e f f e c t i v e l y reduc ing I) o r o f f i n a l s t a t e s (thus reducing C ) . Case 3 : I <_ S < C. This case i s probably 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 f a c t tha t t h i s case p laces 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 rder to so l ve h i s dynamic programming problem i n market M a t date 0 . By knowing the p r o b a b i l i t i e s i^sc} and {ir }, an i n d i v i d u a l can reduce t h i s i n fo rma t ion c o l l e c t i o n problem, by us ing (4.10) and ( 4 . 1 1 ) , to one o f f i n d i n g out C dimensions o f i n fo rma t ion f o r the r c ' s to supplement the . in format ion ( q - | , . . . , q s ) which he can observe i n the date 0 market . However, i f C > S , he s t i l l cannot l e a r n a l l o f the in fo rmat ion he needs un less i t a c t u a l l y has a lower dimension than C. S ince the r ' s are determined by on ly weal th l e v e l s , y ^ , there are FRE's f o r a r b i t r a r i l y l a r g e numbers, C, o f p o s s i b l e f i n a l s t a t e s as long as the number o f c a t e g o r i e s o f agen ts , I, does not become l a r g e r than S , the number o f observab le s i g n a l s ( p r i c e s ) . Un fo r tuna te l y i t i s very hard to get p o s i t i v e o r negat ive r e s u l t s i n t h i s c a s e . F i r s t , here i s a somewhat negat i ve r e s u l t . Counterexample: I f I <_ S < C, there are economies f o r which H = -_ EnF(A-X , GBX)" 1 X- 'G has rank l e s s than I. For example, l e t S = I = 2 , C = 3 , and l e t n-, A-0.-. and, l e t n.-, + ti2 = 0'.. One . may t h i nk o f (m>n2 a s . a vec to r o f • p r o p o r t i o n s - o f weal th r e a l l o -c a t i o n s dy = ( d y 1 , d y 2 ) ' such tha t t o t a l weal th Y = y ^ y 2 i s cons tan t (dY=0). Le t e = ( 5 1 , c 2 , ? 3 ) 1 = - F ( A - X ' G B X ) " 1 X ' G ( n 1 , n 2 ) 1 . I t i s shown i n the Appendix to t h i s chapter t ha t we can s e l e c t n o f f u l l rank such t ha t i t i s a proper c o n d i t i o n a l p r o b a b i l i t y mat r i x and M = 0 (and hence rank H <. 1 < I ) . The counterexample shows tha t i t i s not always the case tha t the s tandard hypotheses o f the i m p l i c i t f u n c t i o n theorem ho ld and hence guarantee the e x i s t e n c e o f a LFRE. I t i s not n e c e s s a r i l y a counterexample to the e x i s t e n c e o f a LFRE. There are many 1-1 mappings which have a s i n g u l a r Jacob ian a t some po in t i n t h e i r domain. For example, l e t f ( x ) = x . S ince f ' ( 0 ) = 0 , the Jacob ian 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 . What i s needed (and has not been found) i s a d i f f e r e n t i a t e path P g iven by y ( t ) = ( y 1 ( t ) , . . . , y I ( t ) ) f o r t e [ 0 , l ] such tha t q ( t ) - q(0) = - / * = 0 E n F ( A - X , G B X ) " 1 X , G d y ( x ) = 0 ( t e [ 0 , l ] ) but r ( t 2 ) - r ( ^ ) = J * | t ( A - X'GBXJ - ^ ' G dy(x) f 0 f o r a lmost a l l t-, < t 2 -There would be no LFRE's anywhere a long t h i s path s i n c e q would be cons tan t a long the pa th , even though r con t i nuous l y v a r i e s . An i n d i v i d u a l cannot t e l l where the economy i s on the pa th , but needs the i n fo rma t i on f o r h i s dynamic program. There are 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 L F R E ' s , however. For example, they do e x i s t . Theorem 4 . 3 : I f , i n economy M ' , the mat r i x o f opt imal ho ld ings X has rank I and I <_ S <_ C, and agent 1 has -1 < Y-| 1 0 , there e x i s t a r b i t r a r i l y - s m a l l - c h a n g e s in {T^}, {TT S C}, {X c } and y 1 such tha t the r e s u l t i n g economy has a LFRE ( i . e . , where q i s l o c a l l y an i n v e r t i b l e f u n c t i o n o f y ) . That i s , the se t o f LFRE economies i s dense in the s e t o f economies o f f u l l rank X m a t r i c e s . Moreover , i f a power u t i l i t y economy M' has rank (H) = I, (and hence i s a LFRE), i t remains a LFRE under a r b i t r a r i l y smal l changes i n the exogenous parameters o f the economy. That i s , the i m p l i c i t f u n c t i o n s u f f i c i e n t c o n d i t i o n f o r e x i s t e n c e o f a LFRE i s an open p rope r t y . Comments: Coupled w i th the counterexample, t h i s theorem suggests t ha t the mat r i x [I^TT^] o f j o i n t i n te rmed ia te and f i n a l s t a t e p r o b a b i l i t i e s i s c r u c i a l i n de termin ing whether enough in fo rma t ion about market M' p r i c e s r can be communicated by date 0 market M p r i c e s q . Th is mat r i x o f 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 t ha t agents may o r may not be ab le to see through a t date 0 . The assumption about the rank o f X w i l l be weakened i n Theorem 4 . 4 . Proof of Theorem 4.3 F i r s t , we shall show that we can perturb the IT'S to ensure rank (H) = I, so that a LFRE eixsts 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 in v e r t i b l e . Recall that H = - EnF(A-X'GBX)"1X'G = - d i a g [ Y s _ 1 ] [ r r s | c ] D where ir^ j = — — — = Pr (intermediate state s | f i n a l state c) c and D = diag[X c](A-X'GBX)" 1X'G Clearly, we can vary the TT s and T T S C SO that the TT s| c 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 of D i s I since the rank of X i s I, and D i s j u s t X pre- and post-multiplied by in 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 of D and suppose w.l.o.g. that rank (D) = I. Also, we may delete rows of [ 7 T S| C] so that we can assume w.l.o.g. that S = I. Then, the determinant of [TT I ]D i s a multinomial i n (TT I S = 1, S j C S j c I - 1; c = 1, ..., C>, say det(Dr s| c]D) = m(irs ,c) . I f we cannot s l i g h t l y perturb the ^ ^  so that rank (H) = I, then m ( 7 r s | c ) E 0 o n s o m e °P e n s e t (in the product of projections of S-l S - simplices into & ). As in the discussion of the counterexample in the Appendix, this can only occur i f a l l the coe f f i c i e n t s of m are i d e n t i c a l l y 0. But, this cannot be so, for i f we define r 1 i f s = c, s l c 1 0 otherwise then m ( T r s | c ) = det(D) f 0 . Hence, f o r some a r b i t r a r i l y smal l pe r -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 . s I c To see tha t the proper ty "rank (H) = I" occurs on an open s e t , note t ha t p r i c e s , demands, e t c . are a l l cont inuous 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 ) and tha t the s e t o f SXI mat r i ces w i th f u l l rank ( i . e . , having submatr ices w i th SI nonzero determinant) i s open i n £ . Now i t on ly remains to be shown tha t the i n v e r t i b i l i t y o f (A-X'GBX) i s an open and dense proper ty i t s e l f . C l e a r l y , i f A-X'GBX i s n o n s i n g u l a r , the behav io r o f the economy i s s u f f i c i e n t l y cont inuous f o r the n o n s i n g u l a r i t y to ho ld under smal l pe r t u rba t i ons o f the parameters o f the economy. I f - 1 < y . < 0 ( i = 1 , . . . . , I ) , , the mat r i x -A + X'GBX has a l l • p o s i t i v e elements and a dominant d iagonal ( c f . G a l e - N i k a i d o [ 1 9 6 5 ] ) 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 CxC mat r i x M has a dominant d iagona l i f there e x i s t <* > 0 (c = 1 , . . . , Q ) such tha t l m cc I a c > * l m cd l a d (c = 1 . . . . . C ) . . d= I d fc In t h i s c a s e , take a = r . c c Suppose tha t - 1 < Y-J ±0, even though o the r agen ts ' exponents may not s a t i s f y t h i s i n e q u a l i t y . By i n c r e a s i n g the weal th y-j and the s o c i a l endowments {X c> i t i s p o s s i b l e to move agent 1 a long h i s Engel curve (a ray through the o r i g i n ) w i thout va ry ing p r i c e s { r c > 9X and o ther agen ts ' a l l o c a t i o n s . Th is occurs i f dX = l_c dy, = x l c dy n L | _____ j ayi y} This has the e f f e c t o f adding more and more o f a dominant d iagona l mat r i x to -A + X'GBX. At some po in t i n the p r o c e s s , the mat r i x -A + X'GBX i t s e l f must assume a dominant d iagonal and become i n v e r t i b l e . Moreover, the mat r i x -A+X'GBX i s l i n e a r i n y- j , p rov ided the X c ' s vary as above so tha t x ^ c i s l i n e a r i n y- j . Hence det (-A + X'GBX) i s a C degree polynomial i n y^ which does not van ish everywhere. S ince i t s zeroes are i s o l a t e d , there must be a r b i t r a r i l y smal l pe r tu rba t i ons o f y-j and ( X c ) tha t make A-X 'GBX 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 , except t ha 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 smal l 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 . } , {X „ } S SC 1 c such tha t a LFRE e x i s t s . Moreover, the economy w i l l be per turbed to a neighbourhood o f i t s exogenous parameters on which rank (X) = J f o r some cons tan t J <_ I, and on which LFRE 's e x i s t . Comment: The proof o f t h i s r e s u l t uses the rank theorem (Dieudonne [1960, p. 273 ] , Rudin [1964, p. 198 ] ) , 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 con -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 of a mapping y -> ze£^ such t h a t r i s an i n v e r t i b l e f u n c t i o n o f z . The techniques o f Theorem 4 .3 . can then be used to ad jus t the j o i n t p r o b a b i l i t i e s { T r s 1 T s c} so tha t p r i c e s q communicate the value o f z and hence r . The important po i n t here i s t ha t i t may not be necessary to s i g n a l a l l the endowment i n f o r m a t i o n , s i nce z i s an adequate summary s t a t i s t i c . I c Rank Theorem: Let y° e N c R , where N i s an open set, and f : N -> & be a continuously d i f f e r e n t i a t e mapping such that, 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 3 onto •1 such that g and g~ are continuously different!'able (here, (-1 .D 1 = {y G &1 in &*), and y i| <1, i = 1 1} i s the open unit ball 2. an open neighbourhood V P f(U) of f(y°) and a function f 1 1 1 h : (-1,1) - ^ 4—V with h and h continuously d i f f e r e n t i a b l e , ' onto J such that f = h 0 f 0 0 g where f ° : (-1 .D 1 -> (-1 ,D C by f ° ( Z ] Zj) = ( z r Z j , 0, 0) H , i ) J f° -(-1,1)1 This theorem says that, i f the mapping f has constant rank J on some open set, then the action of 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-tion theorem. Proof of Theorem 4.4 Theorem 4.4 i s proved by f i r s t perturbing y-j and (X c) to a point where A-X'GBX has f u l l rank, and then perturbing y to some point about which there i s a neighbourhood on which the price 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 result follows by varying TT s| c as in the proof of Theorem 4.3 to ensure that the 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 c) to a neighbourhood on which 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 1 } i s open for any J ' , due to the continuity of the mapping y —> det (X) in market M1 where X i s any square submatrix of X. Since rank (X) i s bounded above by I, we can choose any y° for which the corresponding matrix X° has rank J where J i s the lim sup of the ranks of 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 in the neighbourhood N, since i t i s merely the matrix X1 pre- and post-multiplied by i n v e r t i b l e matrices. Now, apply the Rank Theorem. Using the notation (z.,, Z j , 0, 0) = f°0g(y), one can see that prices r = h(z-|, Z j , 0, ..., 0) = R(z), say, where z = (z-, Z j ) . By (4.13) q = E Fnr = d i a g [ Y s - 1 ] C- s| cl d iag [X c] 0 R (z) where the c o n d i t i o n a l p r o b a b i l i t i e s T T s | c are de f i ned i n the proof o f Theorem 4 . 3 . . ' . M = d iag [Y - 1 ] [ V . ] d iag [ X l — 3 z 1 -ai By c o n s t r u c t i o n , 3 h / 3 z has f u l l 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 the proof o f Theorem 4 .3 to f i n d an a r b i t r a r i l y smal l p e r t u r b a t i o n o f {T S| C} SO t ha t 3 q / 3 z a l s o has f u l l rank J . That i s , q i s l o c a l l y an i n v e r t i b l e f u n c t i o n o f 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 ows them to s e l e c t t h e i r opt imal demand f u n c t i o n s f o r {y. } a t date 0 . is Note tha t agents cannot l e a r n a l l o f the y ' s . They need on ly know the r e l e v a n t i n fo rma t i on impounded i n z . U n f o r t u n a t e l y , z may not have any na tu ra l i n t e r p r e t a t i o n as an observab le economic v a r i a b l e . 4 ^ Q . E . D . Conc lus ion In t h i s c h a p t e r , we have s tud ied whether o r not there e x i s t l o c a l l y f u l l y i n fo rm ing r a t i o n a l expec ta t i ons e q u i l i b r i a (LFRE) i n a power u t i l i t y economy where agents are unce r ta i n about f u tu re p r i c e s because they are unce 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 o f w e a l t h . I t was shown tha t LFRE's " u s u a l l y occu r " i n the. gene r i c sense tha t they occur on an open and dense se t o f parameter!*zat ions 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 his 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 prices. I f there are not more markets than people, then a FRE exists 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 prices. Note that although the analysis of this model used the com-plete market structures M' and M", so that the S markets at date 0 in M w i l l , by Arrow's theorem, allow ultimate allocations as i f the markets were complete i n date 2 goods, the Hirshleifer-Marshall result does not y i e l d the rollover 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 securities of the f i r s t period to the C securities of the second period. The next chapter deals with -constructive models that are not flawed by the rollover strategy. Footnotes to Chapter 4. 1. Openness alone i s not a strong condition, since the unit ball is open in &n, but in many senses is an i n s i g n i f i c a n t part of g.n. Density alone i s not strong since the property may f a i l under small perturbations. A l t e r n a t i v e l y , some writers connote genericity with f u l l Lebesgue measure. Even i f the underlying topology i s the usual metric topology on £ n, neither notion of genericity implies the other. Although sets of f u l l Lebesgue measure are dense, they are not necessarily open. On the other hand, there are open and dense sets of less than f u l l Lebesgue measure. As an example, consider a modi-f i c a t i o n of the construction of the Cantor set obtained by deleting shorter and shorter closed intervals from the unit interval in so that in the l i m i t the remainder (a modified Cantor set) has measure (rather than zero, as for the Cantor set). This set i s closed and no-where dense, so that i t s complement i s open and dense, but of less than f u l l measure, • c f . , Taylor [1973, p. 94]. 2. I f YI = Y2 = = Yj» preferences aggregate and prices are unique by footnote 2, Chapter 3. Hence, prices should also be unique i f the Y do not vary "too much" cross-sectionally. I f prices are not unique, we must assume a l l agents know which rule the Walrasian auctioneer follows in selecting prices. 3. A l t e r n a t i v e l y , one may think of one agent per class, so that agent desires to know only 1-1 other endowments, given that social endowments are true social endowments less his demands. Ess e n t i a l l y , the same re-sults of this chapter hold under these circumstances with obvious modi-f i c a t i o n s , such as replacing I with 1-1. 4. A reasonable conjecture is that, i f a l l agents or classes of agents have different power u t i l i t y exponents, then J = I holds generically. That i s , the set of endowments and probabil i t ies for which rank (X') = I is dense and open in the set of a l l endowments and probabi l i t ies . The conjecture is true for 1 = 2 , for , suppose r agent i trades to x^  e & . Then rank (X 1) < I i f f x^  = ax 2 for some ae £ . That i s , x-j and x 2 l i e on the same ray through the or ig in . Since the Engel curves are rays through the origin with power u t i l i t y that d i f fer only when the powers d i f f e r , i t must be that y-j = Y 2 a n c ' ^l = a ^2 ' 83 . Appendix to Chapter 4 D e t a i l s o f the counterexample p rov id i ng a Jacob ian o f l e s s than f u l l  rank We d e s i r e ir „ > 0 such tha t sc — (4A.1) ^1^11 + 7T21T21 = ^1 (4A.2) ^1^12 + 1 T2 1 T22 = ^2 (4A.3) ^ I^ IS + ^2^23 = ^3 (4A.4) t-}^u + C 2TT 1 2 + 5 3TT 1 3 = 0 (4A.5) 5-j TT2-, + + ^3^23 = ° * where we are g iven {irc}» U s ) and U c>- Reca l l the i n t e r p r e t a t i o n tha t nn- = d y i , so tha t dr =- (A-X 'G B X ) _ 1 X ' G d y and E, = F dr Equat ions (4A.1) - (4A.3) must be s a t i s f i e d f o r the TT s c 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 equat ions (4A.4) and (4A.5) say tha t nFdr = 0 and hence dq = EnFdr = 0 . Now, E s , c V s c 5 c = ^ .c^s 'scV'c^fc = E X d r (by d e f i n i t i o n o f ir ) = z .dy^ (by Wa l ras ' law or (4 .6) and ( 4 . 8 ) ) = 0 Hence equat ions (4A.1) - (4A.5) are redundant , so we may drop ( 4 A . 5 ) , l e a v i n g 4 equat ions i n 6 unknowns. S o l v i n g f o r the o ther v a r i a b l e s i n terms o f TT^ and TT-^ one can see tha t the f e a s i b l e reg ion i s represen ted by C4A.6) (4A.7) 0 <_ T r i T r 1 2 £_ T 2 (4A.8) °£ 7 rlSl 7 rll " 7 r l ? 2 7 r 1 2 ± $3^3 ' assuming w . l . o . g . tha t £ 3 >0. S ince -z-are not a l l o f the same s i g n , so assume w . l . o . g . tha t ^ < 0. S ince TT15 TT2 and £ 3 TT3 are a l l p o s i t i v e , there i s a nonempty open subset N o f the p o s i t i v e quadran t .o f & , hav ing the o r i g i n as a l i m i t p o i n t , which i s conta ined i n the f e a s i b l e se t f o r ( 7 r 1 1 , i r 1 2 ) g iven by (4A.6) and (4A .8 ) . Now, the determinant o f the f i r s t two columns o f n i s a mul t inominal i n i r n and TT 1 2 ( a f t e r s o l v i n g f o r the o ther TT S C'S i n terms o f t h e s e ) , and hence can van ish every Where in N on ly i f i t van ishes everywhere i n & . ( I f i t van ishes i n N, a l l i t s 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 the c o e f f i c i e n t s . ) But s e t t i n g -ni-nn = -Hi and i r 1 2 = 0 (so tha t TT2TT22 7T2) g ives a nonzero determinant o f the f i r s t two columns o f n. Hence one can choose n o f f u l l rank to s a t i s f y (4A.1) to ( 4 A . 5 ) . 2 Chapter 5 P r i c e s Revea l ing Aggregate R isk A v e r s i o n , Impatience and P r o b a b i l i t y B e l i e f s I n t r oduc t i on The two prev ious chapters showed how p r i c e s can revea l aggregate p re fe rence and endowment i n fo rma t ion i n an exchange economy. The o ther major f a c t o r i n f l u e n c i n g p r i c e s i s p r o b a b i l i t y b e l i e f s (when asse ts are r i s k y ) . Th is chapter s t u d i e s the ques t ion o f when p r i c e s revea l p re ferences and p r o b a b i l i t y b e l i e f s . Two types o f models are d i s c u s s e d . One i n v o l v e s a one pe r i od wor ld w i th consumption a t two dates and three s e c u r i t i e s ; cu r ren t con-sumpt ion, 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 tu re consumpt ion. The r e s u l t i n g two r e l a t i v e p r i c e s may revea l aggregate r i s k ave rs i on and p r o b a b i l i t y b e l i e f parameters. The o the r model i nvo l ves two p e r i o d s , so tha t the i n t r o d u c t i o n o f a term s t r u c t u r e i n the bond market c rea tes three r e l a t i v e p r i c e s tha t may revea l an impat ience parameter, i n a d d i t i o n to r i s k ave rs i on and p r o b a b i l i t y parameters . Typ i ca l r a t i o n a l expec ta t i ons models i n the l i t e r a t u r e i n v o l v 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 b e l i e f s o n l y , and the two prev ious chapters 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 i s reasonable to ask why i t should be impor tant to ana lyse the j o i n t r e v e l a t i o n o f these parameters . Pa r t o f the r a t i o n a l e i s r e l a t e d to the m o t i -va t i on f o r i n t e r e s t i n r e v e l a t i o n o f p re ferences a l o n e . Agen ts ' i n f o •mat-ton about .aggregate pre ferences may be : based on econometr ic obse rva t i on o f p r i o r markets (as i n , f o r example, F r i end and Blume [1977] ) , casua l i n f e rences drawn from obse rva t i on o f p r i o r markets , and /o r i n fe rences drawn from obse rva t i on o f c u r r e n t market p r i c e s . Only the l a s t method requ i res r a t i o n a l expec ta t i ons machinery l i k e t ha t developed i n Chapter 3 , and i t i s necessary on ly i f p re fe rences are uns tab le 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 d a t a . The f i nance l i t e r a t u r e has never s e r i o u s l y s tud ied the p o s s i b i l i t y o f uns tab le p r e f e r e n c e s , presumably because researchers 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 asse ts to changes i n be-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 . Without s p e c i f y i n g f u n c t i o n a l forms f o r u t i l i t i e s and s e c u r i t y payof f d i s -t r i b u t i o n s , i t 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 i n fo rma t ion and changes i n pre ferences as s u p e r i o r exp lana t i ons f o r v a r i a t i o n s i n s e c u r i t y p r i c e s . As a r e s u l t , 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 exp la i ned by a sequence o f i n fo rma t ion a r r i v a l s , 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 re fe rences . However, there i s some anomalous ev idence about s e c u r i t y p r i c e s t ha 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 fo rma t ion a r r i v a l s S h i l l e r [1979] and LeRoy and P o r t e r [1979] have ana lyzed the ques t ion o f whether the va r iance ra tes i n t ime s e r i e s o f s tock p r i c e s are too h igh to be exp la i ned s o l e l y by i n fo rma t ion a r r i v a l s . For example S h i l l e r assumes tha t c u r r e n t s tock p r i c e s should be the present va lue o f opt imal f o r e c a s t s o f f u tu re d i v i dend payments. S ince the e r r o r o f an opt imal f o r e c a s t i s or thogonal to the f o r e c a s t , - the va r iance o f an ac tua l d i v i dend s e r i e s should be g rea te r than the va r i ance o f the s tock 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 i n i n t e r e s t r a t e s . This seems to c o n t r a d i c t the hypotheses o f market e f f i c i e n c y and r a t i o n a l e x p e c t a t i o n s , un less one a l s o a s c r i b e s some o f the v a r i a t i o n i n s tock p r i c e s to changes i n p r e f e r e n c e s . Ano ther , somewhat more c a s u a l , obse rva t i on suggest ing tha t p re fe rences may vary over t ime i s prov ided by van Home [1978, pp. 155-161] who notes tha t bond r i s k premia ( i n t e r e s t d i f f e r e n t i a l s between h igh and low grade bonds) vary over the bus iness 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 r i s k . He suggests t ha t i n v e s t o r p re ferences vary over the bus iness c y c l e (people become more r i s k averse i n r e c e s s i o n s ) . Re la ted to van H o m e ' s con ten t i on i s the ques t i on o f whether the suddenness and s e v e r i t y o f the Great Depress ion are r e a l l y adequate ly exp la ined by the t r a d i t i o n a l exp lana t i ons r e l a t e d to bank f a i l u r e s , money supply c o n t r a c t i o n s o r l i q u i d i t y t r a p s . A sudden i nc rease i n r i s k ave rs i on cou ld have p r e c i p i t a t e d the s tock market c o l l a p s e . Thus, there i s mer i t i n s tudy ing the ques t ion o f when v a r i a t i o n s i n p re fe rences 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 i n i n f o r m a t i o n , g iven s p e c i f i c f u n c t i o n a l forms f o r p re fe rences and b e l i e f s . Another f e a t u r e o f t h i s chapter i s t ha t i t p rov ides a c o n s t r u c -t i v e a n a l y s i s o f f u l l y r e v e a l i n g r a t i o n a l expec ta t i ons e q u i l i b r i a (FRE 's ) i n which m u l t i v a r i a t e i n fo rma t ion i s r e v e a l e d . Th is i s i n c o n t r a s t to the l o c a l i m p l i c i t f u n c t i o n r e s u l t s o f the prev ious chapters and the pu re l y t o p o l o g i c a l r e s u l t s o f A l l e n [1978,1979] , tha t e s t a b l i s h gener i c e x i s t e n c e o n l y . The chap te r s t a r t s w i th a general a n a l y s i s f o r exponen t ia l 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 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 progresses to 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 f a m i l i e s : the normal and gamma f a m i l i e s . I t i s a g e n e r a l i z a t i o n o f the a n a l y s i s i n K raus -S i ck [1980] . The c e n t r a l ques t ion o f t h i s chapter i s whether p r i c e s can revea l p r o b a b i l i t y i n f o r m a t i o n , r i s k ave rs i on and impat ience parameters . The aggregate r i s k ave rs i on parameter s t ud ied here i s de r i ved e x p l i c i t l y from i n d i v i d u a l r i s k ave rs i on parameters . Hdwever, on ly a s i n g l e p r o b a b i l i t y parameter f o r a l l agents i s c o n s i d e r e d , a l though under some c i r c u m s t a n c e s , one may p r e f e r to t h ink of i t as an aggregate s u f f i c i e n t s t a t i s t i c o f d i v e r s e i n f o r -ma t ion , as i n Grossman [1976] . A l t e r n a t i v e l y , one may th ink o f the va lue o f the parameter be ing exogenously revea led to some i n -formed agen ts , and communicated to o ther agents by market p r i c e s ( i n a FRE) . S i m i l a r l y market p r i c e s depend on an aggregate impat ience parameter n, which w i th exponent ia l u t i l i t y may be thought o f as a geometr ic mean o f i n d i v i d u a l impat ience parameters , as i n Rub ins te i n [1974] , o r as a commonly he ld impat ience parameter . In the l a t t e r c a s e , one must j u s t i f y why t h i s parameter must be revea led to agents i f they a l ready know t h e i r own personal p r e f e r -ences . 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 agent w i t h an e n t i r e l y un re la ted pre fe rence 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 next pe r iod prices. This i s the same rationale used in Chapter 3, when dis-cussing the need for inference of future prices, in the presence of the rol l o v e r algorithm. Notation and setting The following notation i s s i m i l a r to that used in the two previous chapters: There are I agents ( i = 1 I) y.j t = date t wealth of agent i (t = 0 ,1 ,2) , c» t = date t consumption of agent i (t = 0,1,2) = agent i ' s investment in the date t r i s k l e s s asset that matures at date T ( t , x = 0 ,1 ,2 ; t < x ) m i t = agent i's investment at date t i n the risky asset that pays o f f at date t + l ( t = 0,1) Dt = date t price of the r i s k l e s s bond, maturing at date x ( t , x = 0 ,1 ,2 ; t < x ) . P t = date t price of the risky asset (t = 0,1) Cj. = aggregate date t dividend yielded by risky asset (t = 0 ,1 ,2) . Conditional on information available at any date prior to t , any variable with subscript t i s , in general, random, and this 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 constraints are for dates 0 and 1, respectively: < 5 J ) ^ i 0 = C i O + f i 0 1 D 0 1 + f i 0 2 D 0 2 + m i 0 P 0 (5.2) * i l = c i l + f i l 2 D 1 2 + m i l P l ' Date 1 and date 2 wealth are realized as (5.3) y „ = f i Q 1 + f i 0 2 D 1 2 + m i 0(P 1 +C 1) (5.4) y i 2 = f i l 2 t . ^ Note that (5.3) r e f l e c t s the fact that the date 0 purchase of a two period bond ( f i Q 2 ) y i e l d s f i 0 2 D 1 2 a t d a t e 1 a n d t n a t t n e date 0 purchase of the risky asset (nUg) yie l d s m. 0 ci i n c o n s u m P -tion dividend at date 1 with m^P^ remaining in capital value. A l l wealth i s consumed at date 2, so (5.5) c. 2 = y. 2 . The date 0 market clears when (5.6) The date 1 market clears when (5.7) M = 1 r c o * f i o r 0 f i 0 2 0 • m i o • 1 ' c i l ^ r c i i f i 0 1 = 0 • m i l J 1 N J The following assumption i s maintained throughout: (Al) The stochastic process generating (C Q, C-|, C 2) i s a Markov process with independent increments: A t s C t 't-1 (t = 1, 2) The d i s t r i b u t i o n of i s assumed to depend- on an arbitrary parameter t,^ 91. One-per iod models.: Exponent ia l u t i l i t y Th is s e c t i o n cons ide rs markets t ha t are open a t date 1 o n l y . Assume tha 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 th cons tant abso lu te r i s k ave rs i on f o r con-sumption a t dates 1 and 2 . That i s (5 .8) U n ( c i l f , f i l 2 , m i 1 ; ) = ' - exp ( - 0 ^ ) - n E^expf -e .y^. where y i 2 i s a f u n c t i o n o f f . .^ and m^ by (5.4) and the expec ta t i on opera to r E„ i s f o r the d i s t r i b u t i o n o f C 0 c o n d i t i o n a l on date 1 i n fo rma t ion which i n c l u d e s : , p - j» D^ 2 and C-,. The c o n d i t i o n a l d i s t r i b u t i o n o f C 2 i s a f u n c t i o n o f the parameter r,2 • Two d i s -t r i b u t i o n s are cons ide red he re : normal and gamma. The c o e f f i c i e n t o f absolute^ r i s k ave rs i on i s 0^  > 0 and the ra te o f impat ience i s n, which i s common to a l l agents . Assume (A2) n i s known to a l l agents a t date 1. At date 1, agent i maximizes (5.8) sub jec t to (5 .2 ) and ( 5 . 4 ) . Assuming tha t the o rde r o f the expec ta t i on and d i f f e r e n t i a t i o n opera to rs can be reversed (as w i l l be v e r i f i e d be low) , the f i r s t o rder c o n d i t i o n s f o r agent i ' s demands can be w r i t t e n as i n Chapter 3 , as (5 .9) D ] 2 e x p ( - 0 . c i l ) = n E,;:' exp(-0 i y i 2) (5.10) P ] e x p ( - 0 . c i l ) = n E ^ , C 2 e x p ( - 0 . y . 2 ) . Note tha t the Kuhn-Tucker-Lagrange f i r s t o rder c o n d i t i o n s are necessary 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 i s s t r i c t l y concave and the c o n s t r a i n t s are l i n e a r ( c f . Zangwi l l [1969, c h . 2 ] ) . At this point one may use s p e c i f i c probability distributions for the expectations and compute demand functions (at least in principle) and solve for equilibrium prices with the market clearing 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 bel i e f s and information , the f i r s t order conditions and market clearing conditions are s a t i s f i e d when e/e, (5.11) m i l i l 2 = l T D ^ i l " m i l ( C 1 + P 1 » c i l = f i l 2 + m i l C l (i = 1. I) (5.12) (5.13) P, = nE r C exp(-e(C 2- C J ) J12 nE exp(-e(C 2- C J ) ^ 2 and. where e" 1^^^^" 1) Multiplying (5.13) by C-j and subtracting from (5.12) yi e l d s (5.14) P 1 - C1 D 1 2 = n E ? 2 (C 2-d) exp(-e(C 2-Ci)) Agents desire to i n f e r 5 2 (when e i s unknown, but n> C-| are known) from prices P^ and D-|2 in a rational expectations equilibrium. Recalling that A2H C 2 - C i , i t i s apparent from (5.13) and (5.14) that, by observing equilibrium prices, agents observe f(e-,c 2) ~ E K z exp(-o A2) and. - . f e - - f ( e » £ 2 ) = E c A 2 e x p(-8A2) » assuming, a g a i n , t ha t the o rder o f expec ta t i on and d i f f e r e n t i a t i o n can be r e v e r s e d . As exp la ined i n Chapter 2 and a p p l i e d i n Chapters 3 and 4 , the procedure f o r sea rch ing f o r a f u l l y in fo rming r a t i o n a l expec ta t i ons e q u i l i b r i u m (FRE) w i l l be to assume e and £2 are known to a l l , and then examine under what c o n d i t i o n s the r e v e l a t i o n o f t h i s i n fo rma t ion can be sus ta i ned i n e q u i l i b r i u m . One-per iod model : Normal ly d i s t r i b u t e d re tu rns C o n s i d e r , f i r s t , a normal ly d i s t r i b u t e d A 2 . Suppose (5.15) A 2 * N ( c 2 , a 2 2 ) where a 2 2 i s assumed known. e 2 a 2 Then D 1 2 = n f ( e , c 2 ) = nexp( -ec 2 + - i - 2 - 2 - ) and P i - C i D 1 2 = - n - l l f ( e , * ; 2 ) = nf(e,C2)(?2-e<?2 2) . 3 8 Hence, agents observe 2 2. (5.16) In ( D 1 2 / H ) = - 6 c 2 + and (5.17) P1/D12-C1 = c 2 - e a 2 2 • Given the l e f t hand s i d e s , these are two equat ions i n two unknowns, 8 and c 2 . S u b s t i t u t i n g f o r £2 from (5 .17) i n t o (5 .16) y i e l d s (5.18) In ( n / D 1 2 ) = ( P i / D i 2 - C i ) e + ( a 2 2 / 2 ) e 2 . Case a Case b F igure 5.1 S o l u t i o n o f Equat ion (5.18) The r i g h t hand s i d e of (5.18) i s a -pa rabo la through the o r i g i n t ha t opens upward, as graphed in two p o s s i b l e cases in F igure ( 5 . 1 ) . Cases a and b occur as zero i s the s m a l l e r o r l a r g e r roo t o f the r i g h t hand s i d e o f (5.18) r e s p e c t i v e l y . In e i t h e r c a s e , two roo ts may occur f o r most va lues o f l n ( n / D 1 2 ) , so we must take advantage o f the r e s t r i c t i o n e > 0 . That i s , we r e q u i r e t ha t the aggregate i n v e s t o r be r i s k a v e r s e . In e i t h e r c a s e , i t i s c l e a r t ha t (5.18) p rov ides a unique s o l u t i o n f o r e > 0 i f f e i t h e r ln (n /D 1 2 )>0 o r e = ( C 1 - P 1 / D 1 2 ) / a 2 2 and l n ( n / D 1 2 ) = - ( P 1 / D 1 2 - C 1 ) 2 / ( 2 a 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 not r obus 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 ) . S ince t h i s i s a one per iod economy, uninformed agents are not g e n e r a l l y i n t e r e s t e d in i n f e r r i n g 0. However, having computed 0, they can compute s 2 us ing ( 5 . 1 7 ) . Th is a n a l y s i s y i e l d s the f o l l o w i n g : Theorem 5 . 1 : In the one-per iod economy, o f t h i s s e c t i o n , where a l l agents have cons tan t abso lu te r i s k a v e r s i o n and the same ra te o f impat ience n, and the r i s k y asse t has a normal ly d i s t r i b u t e d payof f w i th i n i t i a l l y unknown mean re tu rn z,2  + Ci> there e x i s t s a FRE whereby a l l agents can i n f e r c,z ( a n c * aggregate r i s k ave rs i on e) from p r i c e s , p rov ided D 1 2 < 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 . I t i s o f i n t e r e s t to exp lo re the meaning o f the c o n d i t i o n D 1 2 < n. I t says t ha t the one pe r iod d iscoun t f a c t o r i s l e s s than the ra te o f impa t ience . From ( 5 . 1 6 ) , t h i s i s e q u i v a l e n t to 2 ?2 _ 6 °2 11 > 0 or E ( C 2 ) - 6 a22/z > C l - That i s , a FRE occurs 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 the date 1 amount o f s o c i a l consumpt ion. (Here , the c e r t a i n t y e q u i v a l e n t i s taken wh i l e i gno r i ng the impat ience parameter , • e f f e c t i v e l y s e t t i n g n = 1.) One-per iod models: Gamma d i s t r i b u t e d re tu rns Suppose A2 has a non-cen t ra l gamma d i s t r i b u t i o n . That i s , suppose the dens i t y o f A2 i s , g iven a,3 (a>0) , (5.19) PB;'.U2> - ( V ( - A , + g ) , ( 5 2 i s ) where r ( « ) i s the incomplete gamma f u n c t i o n which normal izes the p r o b a b i l i t y to i n t e g r a t e to u n i t y . Both a and 3 are cand ida tes f o r the parameter ? 2. B i s a n o n - c e n t r a l i t y parameter s p e c i f y i n g I c o a t i o n o f the d i s t r i b u t i o n . Given a, an i n c r e a s e i n 3 y i e l d s an i nc rease i n the mean increment E (A2) = E ( C 2 ) - C i i n the s o c i a l consumption p rocess . We assume (A3) 3 > - CT_ . Since A 2 = C 2 - >_ g _> - Cj_, t h i s ensures tha t C 2 >_ 0 . Th is overcomes one o f the problems a s s o c i a t e d w i th normal ly d i s t r i b u t e d returns-: they l ack l i m i t e d l i a b i l i t y and ho lders o f the r i s k y asse t may be fo rced to consume a r b i t r a r i l y negat ive amounts o f the consumption good - - an incomprehens ib le t a s k . The mean o f A 2 i s a + 3 and the va r i ance i s a, so a i s s imu l taneous ly a l o c a t i o n and s c a l i n g parameter . Eva lua t i ng the moment genera t ing f u n c t i o n . o f A 2 y i e l d s . f (e,5 2) = E r exp(-e A 2 ) £2 and, 3 ) = exp (-6 3 ) (1+0)" (0>- l ) 3 f ( e , ? 2 ) = E r A 2 exp (-0 A 2 ) 3 0 ' - ? 2 = (3 + 0 1 / ( 1+0)) exp (-03) ( l +e ) " . Here, we s e t c 2 equal to a (or 3) w i th 3 (or a) f i x e d and known. Hence, from ( 5 . 1 3 ) and ( 5 . 1 4 ) , agents observe D 1 2 = n exp ( - 0 3 ) ( l + 0 ) ~ A o r ( 5 . 2 0 ) l n ( D 1 2 / n ) = - 03 - a £n (1+0) and Pi - C 2 D 1 2 = ( 3+ a / ( l + 0 ) ) D 1 2 or ( 5 . 2 1 ) P1/D12 - C i = 3 + a / ( l+e ) . Case 1 1,1 - a;-,3 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 i n t o ( 5 . 2 0 ) y i e l d s ( 5 . 2 2 ) In (n / D 1 2 ) = 03 + ( l + e ) ( P 1 / D 1 2 - C x - . B) l n ( l +e ) . 98. (eio) Figure 5.3 Graphs o f F ( e ) , the r i g h t hand s i de o f equat ion (5.23) 99. The right side of (5.22) i s a convex function through the o r i g i n as in Figure 5.2. Note from (5.21) that Px/D12-C1 -3>0. The right hand side of (5.22) i s s t r i c t l y monotone increasing i f B>0, always y i e l d i n g a FRE (since (5.21).gives a in terms of e). But assuming g>0 places strong conditions on the social consumption process: i t requires that the social consumption levels never decrease from one period to the next. In terms of endogenous variables i t i s clear that e can be inferred (.since e>0) i f f n >Di2> which is the same condition as for Theorem 5,1. Once again we note from (5.20) or (5.13) that this requires that the expected marginal u t i l i t y of date 2 consumption be less than that of date 1 consumption disregarding impatience. Since marginal u t i l i t y i s decreasing in wealth, i t means that the certainty equivalent of date 2 consumption exceeds date 1 consumption. We have established: Theorem 5.2: In a one-period economy as in this section (constant absolute r i s k aversion, known impatience n, non-central gamma d i s t r i b u -ted social payoff increments with known non-central ity. parameter e) , there exists a FRE whereby a l l agents can infer c 2 = a ( a n c l 6) f r o m prices i f f D i 2 s n. Case 2 s 2 = 3; a known Solving (5.21) for £ and substituting into (5.20) yields (5.23) ln(n/Di 2) = e ( P i / D ^ - q )-ae/(l+0 )+a ln(l+e) Let the r i g h t hand side of (5.23) be F(e). Then F(0)=0 and F(e)-> + » or - <*> as e-*- «=, according as P 1/D 1 2-C 1 i s greater than or less than zero, respectively. Now, F'(e) = I V D i 2 - C i ~ (i+e)* W ' so F i s asymptotically concave as e-*», and F' has the same sign at 6 = 0 and e = + oo as Pi/D 1 2-C 1. The zeros of F' are I f P\/D12-Cl>0, there are no positive roots of F' so F i s monotone increasing. If P i / D 1 2 " c i < " " / / 2 ' F' has no roots, so F .is'monotone decreasing. I f -a/2<P1/D12-Cl, there are two positive roots of F (perhaps not d i s t i n c t ) , so F i s decreasing, then increasing, then decreasing as 6 increases. These cases are i l l u s t r a t e d in Figure 5.3. The f i r s t case has a FRE i f f ln(n / D 1 2)>0 and the second has a FRE i f f ln(n / D 1 2)<0. In the t h i r d case, F i s i n v e r t i b l e only for suitably large e. That i s , for 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 for e>e*, or, equivalently, ln(n /D 1 2)<k<0. This y i e l d s : Theorem 5.3: In the one-period economy of this section with non-central gamma distributed social payoff increments there exists a FRE whereby a l l agents can i n f e r the non-centrality parameter s 2 = e (and e), from prices, given a, i f f either i ) n >_ D 1 2 i i ) n < D 1 2 i i i ) n < K D i 2 where 0<K<1 and K i s a function P i , D i 2 , C i , C 0 , n and ot. Note that conditions i ) , i i ) and i i i ) are non-vacuous, for i ) obtains i f ot, e, e>rj, i i ) obtains i f 0< a i s large, 3<_-5a (say) and 0<6 i s close 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 certainty equivalent (disregarding impatience) of date 2 consump-tion exceeds date 1 consumption. The condition Pi-CiDi 2>0, for example, is that the value of the risky asset paying o f f at date 2 should exceed the value of the certain date 1 consumption, i f consumption i s postponed and P 1 / D 1 2 - C 1 >_ 0 or, and P 1 / D 1 2 - C 1 1 - I or, and - I < P f /D i2 -C i '< 0 , to date 2. These c o n d i t i o n s sound very s i m i l a r to each o t h e r , but are mathemat i ca l l y q u i t e d i s t i n c t , as seen i n equat ions (5.20) and ( 5 . 2 1 ) . Cond i t i ons i i ) and i i i ) show t h a t , i n - g e n e r a l , - n > D i 2 i s nei ther": necessary nor s u f f i c i e n t f o r a FRE,. which i s , the case, i n Theorems (5.1) and ( 5 . 2 ) . For general 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 parameters , however, i t i s reasonable to expect tha t the r e l a t i o n s h i p between n and D 1 2 should be impor tant i n determin ing the e x i s t e n c e o f a FRE. From ( 5 . 1 3 ) , D 1 2 = n when e = 0 , and , under reasonable c o n d i t i o n s t h i s e q u a l i t y should be ob ta ined i n the l i m i t as e-> 0 + . (Note tha t f o r e = 0 , the u t i l i t y f u n c t i o n i s a c o n s t a n t , so p r i c e s f o r e = 0 are not w e l l - d e f i n e d . ) Th is w i l l s t i l l be the case i f one so l ves (5.12) f o r z2 as a f u n c t i o n of e ( c o n d i t i o n a l on p r i c e s ) and s u b s t i t u t e s i n t o (5.13) to get an equat ion i n e a l o n e . Th is l a t t e r equat ion w i l l a l -ways be comparing l n ( n / D 1 2 ) to a f u n c t i o n of e t ha t van ishes a t 0 . The f u n c t i o n may have many shapes, as i n F igures 5.1 to 5 . 3 , but i f i t has a t most one s t a t i o n a r y po in t f o r e > 0 , a FRE w i l l e x i s t de-pending on ly on the s i gn o f l n ( n / D 1 2 ) . One pe r iod models: Other d i s t r i b u t i o n s and u t i l i t y c l a s s e s The exponent ia l u t i l i t y c l a s s was used in the prev ious 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 ave rs i on parameters) and because i t o f ten y i e l d s c l o s e d form s o l u t i o n s to the expec ta t i on f ( e , £ 2 ) = E e x p ( - 6 A 2 ) tha t i s used i n equat ions (5.13) and ( 5 . 1 4 ) . For exponent ia l u t i l i t y , one merely needs to assess the moment gene-r a t i n g f u n c t i o n of A2, as was done i n the normal and non-cen t ra l gamma d i s t r i b u t i o n c a s e s . Another approach i n v o l v e s us ing the o the r l i n e a r r i s k t o l e r -ance u t i l i t y f unc t i ons tha t permi t agg rega t i on , extended power and l o g : UTI ( c i i > f i i 2 ' m i i ) = T'^I^II)^"Vv^J* (Of Y <l ) or U n ( C i ! » f i 2 ' m i i ) = l n ( e i + c i l ) + nE l n ( 8 . + y . 2 ) (y=0) Fo l l ow ing the type o f development o f the prev ious s e c t i o n s and o f Chapter 3 , p r i c e s s a t i s f y (5.24) D 1 2 ( C 1 + 0 A ) Y _ 1 = n E ( C 2 + e A ) Y _ 1 (5.25) P1 ( C 1 + e A ) Y _ 1 = n E c 2 C 2 ( C 2 + e A ) Y _ 1 (Y<1) where e A = . Here a g a i n , we have assumed tha t the o rde r o f d i f f e r e n t i a t i o n and expec ta t i on can be r e v e r s e d . The equat ions can be combined to y i e l d (5 .26) ( C 1 + e A ) Y _ 1 ( P 1 + e A D 1 2 ) = n E ? 2 ( C 2 + e A ) Y . In the case o f l og u t i l i t y , y = 0 and (5.26) becomes p i + e A D i 2 = n (C 1 +e A ) which can be so l ved f o r e A p rov ided n|=Di 2. S u b s t i t u t i n g y = 0 i n t o (5 .24) y i e l d s h2 ( C 2 + e A ) = n ( C 1 + e A ) The r i g h t s i de i s known a t date T and the ques t ion 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 . Note tha t (C 2 +e A ) 1 i s a dec reas ing convex f u n c t i o n o f C 2 so tha t i f the parameter z,2 ranks the d i s t r i b u t i o n s i n accordance w i th s t r i c t second degree s t o c h a s t i c dominance ( see , f o r example, Hanoch-Levy [1969]) then i t can be i n f e r r e d from p r i c e s . Thus, f o r example, i f ? 2 i s a l o c a t i o n parameter such as the mean, i t can be i n f e r r e d from p r i c e s and a FRE e x i s t s w i th l og u t i l i t y . 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 s t a t e the assumption o f the i n t e r c h a n g e a b i l i t y o f expec ta t i on and d i f -f e r e n t i a t i o n : (A4) Interchange of 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 . Assume f o r a l l £ 2 t ha t ^ C 2 ( m C 2 + f ) Y - E ? 2 ^ ( m C 2 + f ) Y and W h 2 { ^ f ) y = h2 W^^2+f)y (Ofr< l ) o r t h a t A E C 2 l n ( m C 2 + f ) = E 5 2 . 3 ^ 1 n ( m C 2 + f ) and g | - ' E ln(mC 2 +f) = E g |^ . ln( rnC 2 +f) (y=0) . 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 p r i n c i p l e , 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 ( f am i l y ) and then e i t h e r e x p l i c i t l y d i f f e r e n t i a t i n g o r us ing the Lebesque dominated convergence theorem. C losed form s o l u t i o n s f o r expected l og 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 p r o b a b i l i t y f a m i l i e s are r a r e , so i t i s d i f f i c u l t to p rov ide numerical examples to e s t a b l i s h the v a l i d i t y o f (A4) or i l l u s t r a t e the f o l l o w i n g theorem. Theorem 5.4 Cons ider a one per iod economy as i n t h i s s e c t i o n where the d i s t r i b u t i o n o f date 2 s o c i a l wea l th C 2 i s parameter ized by c 2 . Suppose t ha t ? 2 < ? 2 * i f f C 2 9 e n e r a ' t e c ' by s t o c h a s t i c a l l y dominates (second degree) C 2 generated by ? 2 . Then i f a l l agents have extended log u t i l i t y , a FRE e x i s t s whereby date 1 p r i c e s revea l s 2 and e ,^ i f D 1 2 * r\. Two pe r iod models : Exponent ia l u t i l i t y The s o l u t i o n s to the general model a t date 1, as expressed i n equat ions (5 .9) to (5.14) can be used to o b t a i n a de r i ved u t i l i t y f o r date 1 weal th y ^ , which can 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 no t ing (5 .9) y i e l d s the f o l l o w i n g exp ress ion f o r date 1 de r i ved u t i l i t y f o r ' agent i ( c f . ( 5 . 8 ) ) : - ( l+D 1 2 )exp( -e . C i l ) (5.27) U i l ^ i l > = - (1+Di 2 )exp :y i l :-(e/e i )(Ci+Pi) 1+D 12 + (0/6.)C 1 (5.28) 6 • (1+D12)<j) exp ( - 7 + 0 ^ ^ 1 1 ) w h e r e exp f o ( P i - D i 2 C i ) ) 1+D 1 2 In g e n e r a l , as viewed a t date 0 , <j> i s random, s i n c e P i , D i 2 and Ci a re random. However, us ing assumption ( A l ) , which i s t ha t the increments A ^ = - C t_-j o f the s o c i a l consumption process are independent , equat ion (5.14) shows t ha t P i - D ^ C x i s n o n - s t o c h a s t i c a t date 0 ( i f n and e are known) s i n c e i t depends on ly on the d i s -t r i b u t i o n of A 2 about which agents have the same " in format ion a t date 0 as a t date 1. S i m i l a r l y , i n (5 .13) the va lue o f D 1 2 depends on l y on the d i s t r i b u t i o n o f A 2 i f n and e are known, so tha t i t i s 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 are on ly i n t e r e s t e d i n FRE's i n which date 0 p r i c e s revea l n and 6 (and S i or 5 2 ) . Note tha t the cons tan t abso lu te r i s k ave rs i on (wi th no weal th e f f e c t s on p r i c e s ) and the assumption of a Markov s o c i a l consumption process w i th independent increments combine to m a k e ; D i 2 : ( a n d f u t u r e spot i n t e r e s t r a t e s ) d e t e r m i n i s t i c . Adding i n exponent ia l u t i l i t y f o r date 0 consumption and d i s c o u n t i n g the date 1 de r i ved 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 o f (5 .29) ib1gi0' fi01 ' f i 0 2 ' m i 0 ^ = - e x p ( - 0 c i o ) - T i E ? i ( e x p ( - e i c 1 1 ) + n E ? 2 e x p ( i - e . c . 2 ) ) = - exp(-e.c i 0)-n<(>( l+Di 2 ) e 1+D where y - , s a t i s f i e s ( 5 . 3 ) . S ince dl2 1 S d e t e r m i n i s t i c a t date 0 , the pure expec ta t i ons v e r s i o n 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 es must ho ld ( i n the absence o f a r b i t r a g e ) so tha t (5 .30) D 0 2 = D 0 1 D 1 2 ' Th is a l l ows agents to s e t f . Q 2 = 0 w i thou t l o s s o f g e n e r a l i t y , s i n c e two-per iod r i s k l e s s investments can be obta ined by r o l l i n g over a one-per iod bond a t date 1. Assume CQ i s exogenously revea led to a l l agents a t date 0 . Maximiz ing (5.29) aub jec t to f j 0 2 = 0 and (5.1) and (5.2) y i e l d s demands f o r C ^ Q , f IQ -J , and m^-] . Using (5 .6 ) to c l e a r the market y i e l d s p r i c e s . In e q u i l i b r i u m we have: m _ e. _ e / ( l+D 1 2 : ) m i 0 = eT " eTAtTFTJ^) f i 0 = ( D 0 1 / ( l + D 1 2 r V W V C 0 + P 0 » c i 0 = W 1 * 0 ^ " 1 + m i 0 C 0 p 0 = n < l » E ? I [ ( C 1 + P 1 ) e x p ( - e ( l + D 1 2 ) " 1 ( C 1 + P 1 - (1+D 1 2 )C 0 ))] . D Q 1 = n < ( > E C i [ e x p(-6 ( l+D 1 2)" 1(C 1+P 1 - ( l t D 1 2 ) C b ) ) ] Now, ( 5 . 3 1 ) CT+PT-O+D^C Q = ( P 1 - D - | 2 C 1 ) + ( l + D ^ ) ^ and we have seen tha t p - | - D i 2 C l l s n o n - s ' t o c n a s ' t ' ' c a t d a t e °» a s i s D 1 2 - Using the d e f i n i t i o n (5.28) o f ty, the p r i c e equat ions s i m p l i f y to ' (5.32) P 0 = nE C i [ (Ci+Pi )exp(-9A 1)] (5.33) D Q 1 = n E ? i [ exp(-6A!)] . Note from (5.13) and (5.33) t h a t , i f Ai and A 2 are i d e n t i c a l l y d i s t r i b u t e d ( i . e . , Ci=s 2) i n a d d i t i o n to being independent , D Q 1 =D 1 2 I f t h i s were the case then the date 0 p r i c e s P 0 , D 0 1 and D Q 2 would 2 on ly p rov ide two p ieces o f i n fo rma t i on s i n c e D 0 2 = D 0 1 - (by 5 . 3 0 ) , which i s not enough to f u l l y revea l a l l o f e» n and c,l o r z,2 • Hence we must assume tha t £ 1^2 and i n f e r one o r the o ther proba-b i l i t y parameter o n l y , i n o rder to ob ta i n FRE ' s . I t i s conven ient to use (5.31) and (5.14) to re -exp ress (5.32) PQ = n E ? i [(-n^-f (e ,c 2 )+( l+D 1 2 )(C 0+Ai)) exp(-6A!)] where f( e,?2) = E ? 2 exp(-6A 2) . L e t t i n g g (0 ,Si) = E ? i expt-eAj.), t h i s becomes, a f t e r i n te rchang ing the o rder o f expec ta t i on and d i f f e r e n t i a t i o n , (5 .34) P Q = ( - n 2 ^ f ( e , c 2 ) + n ( l + D 1 2 ) C 0 ) g ( e , ? 1 ) " nO+D 1 2 ) z~ 9 ( 6 , ^ ) 1 0 8 . Two period models: Normally distributed returns Suppose that There are two candidates for the probability parameter that is to be revealed: z,2 [t,1 known) and ^ (s 2 known). Both cases wi l l be analyzed here. The case where £ i = c 2 and D ° t h are unknown (a stationary process for A .^) wi l l not y ie ld a FRE since we have seen that i t forces D 0 1 = D 1 2 , eliminating one dimension from the price information. We have g-.(e,ci) = E ? i exp(-6A 1) 2 = expC-e^+e oi2/2.) and f (e ,c 2 ) = exp(-e? 2 + e 2 cr 2 2 / ,2) , so, from (5.33), (5.34) and (5.16), D 0 i = ng(e 9?!) or (5.35) : ln(D 0 1 /n ) = -e?i +d2al1/z and (5.36) P 0 = C - n 2(-?2 +eo2 2 ) f ( e'S2 ) + n ( l +Di2 ) C 0 ] g(e>Ci) - n(l+D 1 2 ) ( -?i+ea 1 2 )g (e ,?i) = D o i L D ^ ^ + d + D ^ J ^ ^ D i ^ ^ d + D i z J a ^ i e + (1+D 1 2)C 0] . At date 0, agents observe P Q, D q 1 and D Q 2 > or equivalently, from (5.30), they observe P Q, Dg-, and D12. These prices are given by (5.36), (5.35) and (5.16). Eliminating n from (5.35) and (5.16) yields (5.37) l n ( D i 2 / D 0 1 ) = e ( ^ - ? 2 ) + 6 2 ( a 2 2 - a 1 2 ) / 2 . We assume a l l agents i n i t i a l l y know Co and c^ 2 and o 2 2 . Case 1 ? 2 known; d , n, e unknown. Solve (5.36) for z1 in terms of e and substitute into (5.37) to get (5.38) l n ( D 1 2 / D 0 i ) = e C O +D^r^Po /Do i -D ^ - O +DiaKo) -^ ] + 9 2 [ ( a 1 2 - a 2 2 ) / 2 . + ( l + D 1 2 ) _ 1 ( D 1 2 a 2 2 + (1+D 1 2)a 1 2))] = e [ ( l + D 1 2 ) " 1 ( P 0 / D 0 1 - D 1 2 c : 2 - ( l + D 1 2 ) c 0 ) - c : 2 ) + 6 2 [ ( 3 / 2 ) C l 2 +;i/2(D 1 2-l)/(D 1 2+l ) a 2 ] The right hand side of (5.38) i s a parabola that-.opens up or down, according as 3a 1 2+(D 1 2-l)/(D 1 2+l) a 2 i s positive or negative, respectively. As in the analysis of Theorem 5.1 in Figure 5.1, this yields a FRE according as l n ( D i 2 / D 0 i ) i s positive or negative, respectively. This y i e l d s : Theorem 5.5 Consider a two period economy where aggregate con-sumption C^ follows a Markov process with independent, normally distributed increments, as in this section. Agents have exponential u t i l i t y w i th aggregate r i s k ave rs i on 0 and impat ience parameter r ] . I f the second pe r i od mean increment E(A2) = e 2 1 S known, but the f i r s t pe r iod increment t,\ i s unknown, then a FRE e x i s t s whereby date 0 p r i c e s o f the r i s k y asse t and the bond term s t r u c t u r e revea l the va lues o f e,n and s l s f o r e>0, i f f e i t h e r 2 i ) D12 > Doi and 3 a x 2 + ( D 1 2 - l ) / ( D 1 2 + l ) a 2 > 0 or 2 i i ) D12 < D o i and 3a ! 2 + ( D 1 2 - l ) / ( D 1 2 + l ) a 2 < 0. To check tha t c o n d i t i o n s i ) and i i ) are not vacuous, note tha t i ) occurs i f Zi>^z>0 and a 1 2=a 2 2 i s a p p r o p r i a t e l y l a r g e , and i i ) occurs i f c 2 i s l a r g e and a 2 2 and n are smal l so tha t . D i ' 2 « T and £ i < c 2 , a i 2 > a 2 2 , f o r a g iven 0 > 0 . Case 2 z\ known; ^2> n> e unknown. So lve (5.36) f o r <;2 and s u b s t i t u t e i n t o (5.37) to get (5.39) l n ( D 1 2 / D 0 1 ) = 0 ( c 1 - P o / ( D O i D i 2 ) - ( l + D 1 2 " 1 ) ( c 1 +Co) ) - 6 2 ( a 2 2 / 2 + ( 3 / 2 + D 1 2 " 1 ) a 1 2 ) The r ight hand, s i de i s a downward opening parabo la from which e>0 : can be i n f e r r e d i f f D 1 2 < D 0 1 . Th is y i e l d s : Theorem 5.6 In a two pe r iod economy as i n Theorem 5 . 5 , except t ha t the f i r s t pe r i od increment Z\ i s known, and s 2 , r\ and 0 are unknown, a FRE e x i s t s t ha t r evea l s these parameters i f f D 1 2 < D 0 1 . I t i s o f i n t e r e s t to study the c o n d i t i o n D 1 2 < D Q I . Th is says t ha t the one pe r i od forward ra te o f i n t e r e s t should exceed the spot r a te o f i n t e r e s t , o r , e q u i v a l e n t l y , t ha t the term s t r u c t u r e o f i n t e r e s t ra tes should be r i s i n g . I l l , Two period models: Gamma distributed returns In this section, the increments to social wealth, A 2 and Ai both follow the non-central gamma distribution 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 2 are ( A 1 - B 1 ) a i _ 1 e x p ( - A 1 + 6 1 ) P«l,&l (Ai) = (Ai>Bi) r M ( A 2 - 3 2 ) 2 e x p ( - A 2 + 3 2 ) and P a 2 , 3 2 ( A 2 ) = ~ ( A 2 > B 2 ) r ( a 2 ) Hence f ( e » s 2 ) = E . exp(-6A2) = exp(-eB 2 ) ( l+e)" a 2 and g(e,ci) = E ? exp(-GAi) = exp(-6B 1 )( l+e)" a i so that, from (5.33), (5.34) and (5.20), (5.40) DQI = n g(e 2 Ci) or ln (D 0 1 /n ) = -93 1-a 1 ln(l+e) and (5.41) P 0 = [ n ( 3 2 W ' ( 1 + e ) ) f ( O i ? 2 ) + ( ; 1 + D i 2 ) c o 3 n g ( 9 1 c i ) ' + n ( l+D 1 2 ) [3 1 + a i / ( l+e) ]g(e, C l ) = D 0 1 [ D 1 2 ( 3 2 + a 2 / ( l + . e ) ) + ( l+Di 2 )C 0 +( l+D 1 2 ) (g 1 +a 1 / ( l+e) ) ] Equation (5.20) becomes, in this notation, (5.42) ln (D 1 2 /n ) = ln (E r exp(-8A2)) = -eg 2 - a 2 l n ( l+e ) Subtracting .(5.'40) from (5.42) yields (5.43) l n ( D 1 2 / D c l ) = e (g 1 -3 2 )+ ln( l+e ) (a 1 -a 2 ) . These are four possible candidates for the unknown probability parameter: a l 9 a 2 , p x and g 2 . This.yields four cases to analyze where one parameter is unknown (along with e and n) and the other three are known (along with C 0 ) . The cases wil l be analyzed and summarized in one theorem. Case 1 t,\ = ai unknown; a 2 , g l s g 2 , known. Solving (5.41) for a and substituting into (5.43) yields By (5.41), the factor in square brackets i s , in equilibrium so that the right hand side is a~convex function of e which vanishes at the origin and is increasing for large e . However, i f 3i<6 2, the function may be decreasing for small e. The analysis is the same as for Case 1 in the one-period gamma distributed returns model (Theorem 5.2) as i l lustrated in Figure 5.2, so that a FRE exists i f f ln(D 1 2 /D 0 i )>0. l n ( D 1 2 / D 0 1 ) = e ( 3 1 - B 2 ) D i 2 a 2 + ( l + D i 2 ) o i > 0,.-T+e 113. Figure 5.4 Graphs of F(e), the right hand side of (5.44) Case 2 c 2 = a2 unknown; , a l s 3 i , B 2 known S o l v i n g (5.41) f o r a 2 / a n d s u b s t i t u t i n g i n t o (5.43) y i e l d s A g a i n , the term i n square brackets i s p o s i t i v e , and the r i g h t 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 e i f 3i>B2)> al though i t i s not n e c e s s a r i l y convex f o r a l l 9>0. (For example, s m a l l , the r i g h t s i d e o f (5.44) i s hot convex. ) Denoting the r i g h t s i de of (5.44) by F ( 0 ) , th ree p o s s i b l e graphs of F(9) are i l l u s t r a t e d i n F igure ( 5 . 4 ) . I f graph.: F i ( e ) or Fs(9) o c c u r s , l n ( D i 2 / D o i ) > 0 i s necessary and s u f f i c i e n t f o r a FRE. I f F2(9) o c c u r s , then there e x i s t s a k> 0 and 0*>O such tha t F (e* ) = k, F i s monotone on [9*,°°) and the i nve rse image o f [k , 0 0 ) under F i s [ e * , ~ ) . That i s , a FRE e x i s t s i f f l n ( D 1 2 / D 0 i ) > k > 0 . Note t ha t k depends, i n general on the known or observed parameters Po> D Q I , D]_2, CO, 04, g l 9 . B 2 . Case 3 Ci = Bi unknown, a i , a 2 , g 2 known. Using the techniques o f Case 1 and Case 2 y i e l d s the equat ion (5.44) l n ( D 1 2 / D 0 i ) = 9 (B 1 -g 2 . ) i f ( 2 + D i 2~ 1)a 1 i s l a rge and the f a c t o r i n square brackets i s (5 .45) ln (D i 2 /Do i ) = 1+Di 2 0 - ( l + D 1 2 ) C o - ( l + 2 D i 2 ) B 2 ] 115. F igu re 5.5 Graphs o f ; G ( e ) , the r i g h t hand s i de o f (5.45) f o r a i > a 2 Figure 5.6 Graphs o f G ( e ) , the r i g h t hand s i de o f (5.45) f o r ai<a2 L e t t i n g Y be the 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 . 4 1 ) , i t must be tha t Y = n+Di2 ) (B i -B 2 )+ (D i2a2+( l+D i2 )on ) / ( l+e) . Th is may be p o s t i v e o r n e g a t i v e , but tends to be p o s i t i v e i n s o f a r as « i , a 2 >0 . Thus, i f the r i g h t s i d e o f (5.45) i s G(e), i t may be tha t G(e) ±°° as e » . A l s o , f o r 0>O, the second term o f G(e) i s convex d e c r e a s i n g , and the 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 , acco rd ing as ai>a2 o r a 1 < a 2 , r e s p e c t i v e l y . S ince the second term i s bounded, G(0) i s convex or concave f o r l a r g e 0 acco rd ing as the t h i r d term i s convex o r concave, a l though G may have any curva tu re f o r smal l p o s i t i v e 0 i f the second and t h i r d terms have d i f f e r e n t c u r v a t u r e . F igures 5.5 and 5.6 i l l u s t r a t e p o s s i b l e graphs o f G(0). In F i gu re 5 .5 where a 1 > a 2 , G i s a s y m p t o t i c a l l y concave as 6 and i n c r e a s i n g o r dec reas ing as Y i s p o s i t i v e or n e g a t i v e . Thus, i f ai>c<2. a FRE e x i s t s i f Y>0 and ln(Di2 /Doi )>0 or i f Y<0 and l n ( D i 2 / D r j i ) < k / < 0 where k i s a f u n c t i o n o f P l s Doi, D i 2 , Co, <*i, a 2 and B 2 . In F igu re 5.5 where a ! < a 2 , G i s g l o b a l l y concave ( f o r 0>O). In t h i s c a s e , a FRE e x i s t s i f f Y>0 and ln (Di 2 /D 0 i )>0 or y<0 and ln (D i 2 /D 0 i )<0 . Case 4 S 2 = $ 2 unknown; 04 , a 2 i , 3 2 known. E l i m i n a t i n g &z from (5.41) and (5.43) y i e l d s ln (D i2 /Doi ) = Dirti^ T -( 1 + .Pi2)Co-3i]- 1^-{a2+(l+Di 2" 1>i} • + l n ( l+0 ) ( a 1 - a 2 ) Unknown P r o b a b i l i t y Parameter Cond i t i ons f o r FRE ° i D12 > D 0 i 0-2 D12 > K2D01 a 1 > a 2 » Y>0 and D 1 2 > D 0 i ai>a2> y<0 and D 1 2<K 3 a 1 < a 2 . Y>0 and D 1 2>DQI a 1 <a 2 . Y<0 a n d D 1 2 < D o i 3 2 a!>a2» K>0 and D 1 2 > D 0 i a ! > a 2 ) K<0 and D 1 2<K ) +D a 1 < a 2 » K>0 and D 1 2>D 0 1 a2<ci2> K<0 and D 1 2 < D 0 i Note: K 2 >1; K 2 depends upon 0 4 , 3 l 9 f32» Doi» D 1 2 » C 0 0<K 3<1; K 3 depends upon ot]_> C42> 32» D 1 2 J C 0 0<K1+<1; K 4 depends upon ot 19 a2> 3 i > p0> Doi» D 1 2> C 0 Y = P0/D01 - ( 1+D 1 2 )C 0 - ( i + 2 D 1 2 ) e 2 < = PQ/DOI - ( 1 + D 1 2 ) C 0 Table 5.1 Cond i t i ons f o r a FRE i n Theorem 5.7 118 L e t t i n g K be the 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 1 2 ( B 2 + B 1 ) + ( D 1 2 o 2 + ( l + D 1 2 ) a 1 ) / ( l 4 e ) , and t h i s may be p o s i t i v e or nega t i ve . 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 th K r e p l a c i n g y. These cases may be summarized i n the f o l l o w i n g theorem. Theorem 5 . 7 : Cons ider the two per iod exponent ia l u t i l i t y economy where the s o c i a l weal th s t o c h a s t i c process has independent non-cen t ra l gamma d i s t r i b u t e d -increments as in t h i s s e c t i o n . At date 0 , agents observe p r i c e s P 0 , D 0 1 , D 0 2 (and hence D 1 2 ) , and attempt to i n f e r e, n and one o f a x , a 2 , 3 : , e 2 , which are p r o b a b i l i t y parameters. Then FRE's occur under the c o n d i t i o n s i l l u s t r a t e d i n Table 5 . 1 . Two per iod models : P o r t f o l i o r o l l o v e r s In a l l o f these models , agents have the same ho ld ings o f the r i s k y a s s e t a t date 1 as a t date 0 . That i s , m^Q = e/e^ = m ^ . Th is suggests t ha 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 technique might make i t un-necessary to i n f e r the t as te and p r o b a b i l i t y parameters , as i n "Chapter 3. In t h i s c a s e , however, agents w i l l always d e s i r e to know something about the p r o b a b i l i t y parameter , s i n c e i t a f f e c t s t h e i r u t i l i t i e s f o r f i n a l (and perhaps in te rmed ia te ) w e a l t h . In the 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 parameters and a l l r ece i ved the same updated in fo rma t ion a t date 1, so tha t the c o n t r a c t curve d i d not s h i f t from date 0 to date 1. In these models , i n fo rma t ion about the p r o b a b i l i t y parameter i s a v a i l a b l e a t date 0 , so tha t a l l agents w i l l 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 . I f some agents wa i t u n t i l date 1 to use the i n f o r m a t i o n , the c o n t r a c t curve w i l l s h i f t adve rse l y f o r - t h o s e agen ts , .and any r o l l o v e r a l go r i t hm must f a i l Conc lus ion In t h i s c h a p t e r , one and two pe r i od exchange economy models were s tud ied where agents had i n t e r t e m p o r a l l y a d d i t i v e exponent ia l u t i l i t y , d i scoun ted by an impat ience parameter n • P r i c e s depend on n, an aggregate r i s k ave rs i on parameter e and a p r o b a b i l i t y parameter . Cond i t i ons are d e r i v e d , tha t are v e r i f i a b l e because they i n v o l v e observab le p r i c e s and known parameters , which ensure the e x i s t e n c e o f a f u l l y in fo rming r a t i o n a l expec ta t i ons e q u i l i b r i u m (FRE) . In a FRE, agents can i n f e r 6 and a p r o b a b i l i t y 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 one-pe r iod model , 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 . Cond i t i ons f o r a FRE i n c l u d e , i n t e r a l i a , the r e l a t i o n s h i p between n and the p r i c e o f a d i scoun t bond, i n the one-pe r iod models and the s lope 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 two-pe r i od models. The models used normal ly d i s t r i b u t e d and non-c e n t r a l gamma d i s t r i b u t e d increments to s o c i a l w e a l t h . Another one-pe r iod model , was found to have a FRE, i n which aggregate 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 parameter c 2 cou ld be i n f e r r e d from p r i c e s , i f the p r o b a b i l i t y parameter ranked the f a m i l y o f d i s t r i b u t i o n s i n the same order as would s t r i c t second degree s t o c h a s t i c dominance. Agents i n t ha t model have extended log u t i l i t y . Footnotes to Chapter 5 1. Equilibrium prices are unique because they can be derived by por t f o l i o separation from a complete market structure (with a continuum of markets) in which prices are unique because of the aggregation properties of exponential u t i l i t y . (Cf., Footnote 2, Chapter 3). 2. To v e r i f y the interchange of d i f f e r e n t i a t i o n and expectation, one could either try to fi n d a measurable bound for the d i f f e r -ence quotients and use the Lebesgue dominated convergence theorem (c f . , Rudin [1964, p. 246]) or actually perform the integration 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 for a variety of u t i l i t y f a m i l i e s , i t i s not clear that i t works with exponential u t i l i t y . If A ^  N(y,c 2) then E(exp( YA)) exp( Y 2a 2/2+y Y) and dj YEexp( YA) ( Ya 2+y )exp( Y V/2+y Y) On the other hand, E(-r- e x p ( Y A ) ) = E A e x p ( Y A ) A e x p ( Y A - ( A - y ) 2 / ( 2 a 2 ) ) dA s exp( Yy+a 2 Y 2/2) f ~ A e x p ( - ( A - ( y + a 2 Y ) ) 2 / ( 2 c 2 ) ) dA = [exp( Yy+a 2 Y 2/2)](y+a 2 Y) , 1 2 1 . since the l a s t intergration i s the same as that performed to calculate the mean of a normal variate. Thus, the order of d i f f e r e n t i a t i o n (w.r.t. coe f f i c i e n t s of A 2 ) and expectation 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 in exp(- e(mA+f)) presents no problem since the factor exp(-of) may be taken outside the expectation. To ver i f y t h e - v a l i d i t y of the interchange of d i f f e r e n t i a t i o n and expectation, suppose A i s a central gamma variate with density A a"exp(-A)/r(a) (A>O) . The results when ^ i s a non-central gamma variate follow.by translating a central variate. Then 9> .- / 7 \ 3/-, \ - a f, \-a-I — Eexp(-yA) = —( 1 + Y ) = -a(l+y) dy 9Y (1+Y>0) . On the other hand, E-^-exp(-YA) = E(-A exp(-YA)) -3y = - ( r ( a ) ) -, 1 '• a - 1 Aexp(-yA)A exp(-A)dA 0 = - ( r ( a ) - 1 ( Y + l ) - a - 1 ( ( Y + l ) A ) a e x p ( - ( Y + l ) A ) d [ ( Y + l ) A ] ^ 0 = - ( Y + l ) " a " ] r ( a + l ) / r ( a ) = - a ( Y + l ) " a _ 1 This j u s t i f i e s ^ t h e interchange of expectation and d i f f e r e n t i a t i o n . 1 2 2 . 4 . I t 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 marginal r a tes o f s u b s t i t u t i o n f o r the aggregate i n v e s t o r . The marginal ra te o f s u b s t i t u t i o n between a propor t iona l c l a i m ' to date 1 and 2 consumption and incrementa l consumption a t date 0 f o r the aggregate i n v e s t o r i s : QTTE Er [C exp(-6C )+nC exp(-GC )] eexp(-eCo) = nE [ C i e x p ( - e ( C i - C 0 ) ) + n E r C 2 e x p ( - e ( C 2 - C 0 ) ) ] = nE C iexp ( -eAi )+ n 2 'E r -exp. ( -e(C 1 - .Co . ) )E r C z e x p ( - e ( C 2 - C i ) ) i Si. - s 2 2 ~ = nE Giexp(-0A 1)+n E exp(-6Ai)E . : c 2 exp ( -e (a 2 ) ) Si ?i Q2 = nE^ i Ciexp( -eA 1 )+nP 1 E j , i exp(-6Ai) us ing (5.12) . Th is i s the same as the exp ress ion (5.32) f o r P 0 . The marginal 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 ( inc rementa l ) consumption a t date 1 and ( inc rementa l ) consumption a t date 0 i s e nE e x p ( - e C i ) — = = nE exp(-QAl) , eexp ( -eC 0 ) 5 1 which i s the same,as (5.33) f o r D 0 1 . Chapter 6 Concluding Remarks Several questions have been asked, and at least p a r t i a l l y answered in this thesis. They include: What i s a rational expectations equilibrium? Drawing upon definitions already in the economics l i t e r a t u r e , information i s defined, in Chapter 2, in terms of a-algebras or partitions of the states of nature, upon which agents may form conditional expectations. Information i s distinguished from b e l i e f s , the l a t t e r being an agents' view of the probability 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 stress the difference between information and beliefs in the theory of rational expectations. Macro-economic models tend to stress the importance of agents forming "correct" beliefs in rational expectations models. Micro-economic models, l i k e those in this paper, tend to concentrate on the information aspects in the theory of rational expectations, while merely making convenient assumptions about b e l i e f s . A rational expectations equilibrium i s characterized as a fixed point in a function space of price random variables. Existence of equilibrium i s affected by at least two sorts of problems: discontinuities in demands induced by discontinuities in information (the continuity problem), and the requirement that agents' excess demands, which may be measurable with respect to different a-algebras, must sum to a non-random constant in equilibrium (the measurability problem). Heuristic arguments advanced here, . concern ing the m e a s u r a b i l i t y prob lem, suggest tha t the e x i s t e n c e o f r a t i o n a l expec ta t i ons e q u i l i b r i a t ha t are not f u l l y in fo rming ( i e . , i n which a l l agents do not have the same in fo rma t ion a f t e r they observe p r i c e s ) may be a ra re occu r rence . An i n t e r e s t i n g ques t ion f o r f u tu re research would be to study the robustness o f the e x i s t e n c e o f e q u i l i b r i u m i n va r ious n o n - f u l l y in fo rming models i n the l i t e r a t u r e , such as F u t i a [1978] , Grossman [1977] and Kreps [1977] . Do p r i c e s revea l i n fo rmat ion about p re fe rences? When the i n fo rma t ion i s about an aggregate p re fe rence parameter i n the u t i l i t y c l a s s e s tha t aggregate (extended power and l o g , and e x p o n e n t i a l ) , Chapter 3 shows tha t the answer i s " y e s " : there e x i s t s a f u l l y in fo rming r a t i o n a l expec ta t ions e q u i l i b r i u m (FRE) . The models there have a two pe r i od s t a t e pre ference s e t t i n g , so the r e s u l t s app ly to a r b i t r a r y p r o b a b i l i t y d i s t r i b u t i o n s . In t roduc ing in te rmed ia te pe r iod consumption does not prevent p r i c e s from r e v e a l i n g the aggregate pre ference parameter , but i t i s hard to model the problem i n such a way tha t agents know which p r i c e s (or combinat ion 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 . In t roduc ing random in te rmed ia te pe r iod l a b o r income, which i s independent o f which s t a t e occurs and i s i n i t i a l l y revea led to the agent who w i l l earn i t , does not prevent p r i c e s from r e v e a l i n g the s a l i e n t pa r t o f aggregate p r e f e r e n c e s , as long as t o t a l ( s o c i a l ) l abo r income i s non-random. I f t o t a l l a b o r income i s random, an ex t ra dimension o f no ise i s added and a FRE does not e x i s t . However, a n o n - f u l l y in fo rming r a t i o n a l expec ta t i ons 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 numer i ca l l y by computer. Do p r i c e s revea l i n fo rma t ion about the d i s t r i b u t i o n o f agen ts ' endowments? Th is i s a m u l t i v a r i a t e problem, w i th a vec to r o f p r i c e s and a v e c t o r o f endowments. In a .two p e r i o d , complete market s e t t i n g where there a re a t l e a s t as many in te rmed ia te s t a t e s (and hence p r i c e s ) as agen ts , and p r o b a b i l i t i e s are a p p r o p r i a t e l y non-degenerate , Chapter 4 shows tha t the answer i s g e n e r i c a l l y " y e s " , p rov ided tha t one i s on ly i n t e r e s t e d i n the e x i s t e n c e of a l o c a l l y f u l l y in fo rming r a t i o n a l expec ta t i ons e q u i l i b r i u m (LFRE) . Compared to o ther models i n the l i t e r a t u r e , 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 echn iques , f o r the p r i c e o f g e t t i n g LFRE 's i n s tead of, F R E ' s . Do p r i c e s revea l and d i s t i n g u i s h between pre ference and p r o b a b i l i t y  parameter i n fo rma t ion? This leads to a c l a s s o f m u l t i v a r i a t e problems tha t can be s tud ied by more c o n s t r u c t i v e techniques than were used i n Chapter 4 , s i n c e the problems are two and three d i m e n s i o n a l . In o rde r to model i n fo rma t ion about p r o b a b i l i t y p a r a m e t e r s , : m Chapter 5 , s p e c i f i c f a m i l i e s o f p r o b a b i l i t y d i s t r i b u t i o n s are used - -normal and non -cen t ra l gamma. P r o b a b i l i t y parameters t ha t are s tud ied are l o c a t i o n and l o c a t i o n - s c a l e parameters . One and two pe r iod models are cons ide red where markets c o n s i s t o f bonds and a r i s k y a s s e t tha t pays consumption d i v idends a t i n i t i a l , i n te rmed ia te and f i n a l d a t e s . The one per iod markets have two r e l a t i v e p r i c e s and c o n d i t i o n s are found under which these p r i c e s revea l pre ference and p r o b a b i l i t y i n f o r m a t i o n . One c o n d i t i o n i nvo l ves the r e l a t i o n s h i p between bond prices (discount factors) and agents' common rate of impatience, where the exponential u t i l i t y i s intertemporally additive, discounted by the impatience factor. Equivalently, this condition involves the relationship between current social consumption and the un-discounted certainty equivalent of random next-date social consumption. Another condition i s the relationship between the value of next-date social consumption, i f deferred one period, as well as various technical conditions that do not seem to be amenable to simple economic interpretation. In the two period models, a term structure of interest rates adds another (bond) price, so that aggregate r i s k preference, impatience and probability parameters may be revealed by prices. Conditions for the existence of a FRE.involve relationships l i k e those in the one period model, as well as the slope of the term structure of interest rates. The techniques used in Chapter 5 are quite general and may be extended to other u t i l i t y classes that aggregate, and other probability f a m i l i e s , since the constructive analysis involves assessing moment generating functions (for exponential u t i l i t y ) or non-central moments (for extended log and power) of the probability families. A simple one period extension to extended log u t i l i t y i s developed, where a FRE exists with any probability family, as long as the probability parameter ranks the family in the same order as second degree stochastic dominance. Conc lus ion Th is 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 n o n - c o n s t r u c t i v e techniques to study the ex i s t ence of a f u l l y in fo rming r a t i o n a l expec ta t i ons e q u i l i b r i u m (FRE) i n one and two per iod models , where p r i c e s convey i n fo rma t ion about p r e f e r e n c e s , endowments and p r o b a b i l i t i e s . Some o f the techniques may be extended to o ther s e t t i n g s . B i b l i o g r a p h y A k e r l o f , George A. [1970] . "The market f o r ' l e m o n s ' : Q u a l i t y unce r t a i n t y and the market mechanism," Qua r te r l y Journa l o f Economics, LXXXIV, #3, 488-500. A l l e n , B. [1978] . "Gener ic ex i s t ence of comple te ly r e v e a l i n g e q u i l i b r i a f o r economies w i th u n c e r t a i n t y when p r i c e s convey i n f o r m a t i o n , " Mimeo., Department o f Economics, U n i v e r s i t y o f P e n n s y l v a n i a . [1979] . . " S t r i c t r a t i o n a l expec ta t i ons e q u i l i b r i a w i th d i f f u s e h e s s . " CARESS Working Paper #79-08, Department o f Economics, U n i v e r s i t y o f P e n n s y l v a n i a . . [1980] . "Expec ta t i ons e q u i l i b r i a w i th d i spe rsed in fo rmat ion e x i s t e n c e , approximate r a t i o n a l i t y , and convergence i n a model w i t h a continuum o f agents and f i n i t e l y many s t a t e s o f the w o r l d . " CARESS Working Paper #80-07, Department o f Economics, U n i v e r s i t y o f P e n n s y l v a n i a . Ar row, K. J . [1953] . "Le r o l e des va leu rs bou rs i e res pour l a s r e p a r t i t i o n l a . m e i l l e u r e des r i s q u e s , " Econometr ie . P a r i s : Center Na t iona l de l a recherche s c i e n t i f i q u e , 41-48. E n g l i s h t r a n s l a t i o n in Arrow [1964] . . [1964] , " the r o l e o f s e c u r i t i e s i n the opt imal a l l o c a t i o n o f r i s k - b e a r i n g , " Review o f Economic S t u d i e s , 31 , 91-96. . [1970 ] . Essays i n the Theory o f R i s k - B e a r i n g . Ch icago : Markham. and F. H. Hahn [1971] , General Compet i t i ve A n a l y s i s . San F r a n c i s c o : Hoi den-Day. B l a c k , F. [1972] . " C a p i t a l market e q u i l i b r i u m w i th r e s t r i c t e d bo r row ing , " Journa l o f B u s i n e s s , 4 5 , 444-454. Brennan, M. J . and A. Kraus [1978] . "Necessary c o n d i t i o n s f o r aggregat ion i n s e c u r i t i e s marke ts , " Journa l o f F i n a n c i a l and  Q u a n t i t a t i v e A n a l y s i s , XI I I #3, 407-418. C a s s , D. and J . S t i g l i t z [1970] . "The s t r u c t u r e o f i n v e s t o r p re -ferences and a s s e t r e t u r n s , and s e p a r a b i l i t y i n p o r t f o l i o a l l o c a t i o n : A c o n t r i b u t i o n to the pure theory o f mutual f u n d s , " Journa l o f Economic Theory, 2, 122-160. Cox, John C , Jonathan E. I n g e r s o l l , J r . and Stephen A. Ross [1978] . "A theory 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 , " Research Paper No. 468, GSB Stan fo rd U n i v e r s i t y , August . Debreu, G. [1959] . Theory o f V a l u e . New York : W i l e y . Dieudonne, J . [ I 960 ] . Foundat ions o f Modern A n a l y s i s . New York : Academic P r e s s . Dreze , Jacques H. [1970] . "Market a l l o c a t i o n under u n c e r t a i n t y , " European Economic Review, Win te r , 1970-71, 133-165. Dybv ig , P h i l i p H. [1979] . "Recover ing a d d i t i v e u t i l i t y f u n c t i o n s . " Ph.D. T h e s i s , Ya le U n i v e r s i t y , Department o f Economics. Dybv ig , P. H. and H. Polemarchakis [1979] . "Recover ing c a r d i n a l u t i l i t y f u n c t i o n s , " unpubl ished manuscr ip t . F e l l e r , W. [1966] . An In t roduc t i on to P r o b a b i l i t y Theory and I ts  A p p l i c a t i o n s . V o l . I I . New York : W i l ey . F r i e n d , I rwin and Marsha l l Blume [1977] . "The demand f o r r i s k y a s s e t s , " i n R isk and Return i n F i nance , V o l . I, I. F r i end and J . L. B i c k s l e r , eds . Massachuset ts (Cambridge): B a l l i n g e r , 101-139. F u t i a , Ca r l A . [1978] . "Ra t i ona l expec ta t i ons i n s p e c u l a t i v e mar-k e t s , " Mimeo., New Jersey (Murray H i l l ) : B e l l L a b o r a t o r i e s . G a l e , D. and H. N ika ido [1965] . "The Jacoba in mat r i x and g loba l un iva lence o f mappings," Mathematische Anna len , 159, 81 -93 . Green, J . [1973] . " I n fo rma t i on , e f f i c i e n c y , and e q u i l i b r i u m , " D i s c u s s i o n Paper 284, Harvard I n s t i t u t e o f Economic Research . Green, J e r r y [1977] . "The non-ex is tence of i n f o rma t i ona l e q u i l i - • b r i u m , " Review of Economic S t u d i e s , XLIV(3) #1.38, 451-464. Green, J . , L. Lau and H. Polemarchakis [1979] . " I d e n t i f i a b i l i t y o f the von Neumann-Morgenstern u t i l i t y f u n c t i o n from a s s e t demands." Forthcoming i n J . Green, e d . , General E q u i l i b r i u m , Growth, and  Trade: Essays in Honor of L i one l McKenz ie . .New York : Academic P r e s s . . Grossman, S . J . [1976] . "On the e f f i c i e n c y o f compe t i t i ve s tock markets where t rade rs have d i v e r s e i n f o r m a t i o n , " Journa l o f  F inance , XXXI #2, 573-585. Grossman, S. J . [1977] . "The ex i s t ence o f f u tu res marke ts , no isy r a t i o n a l expec ta t i ons and i n fo rma t i ona l e x t e r n a l i t i e s , " Review  o f Economic S t u d i e s , XLIV #138, 431-449. and J . E. S t i g l i t z [1975] . "On the i m p o s s i b i l i t y o f in fo rma-t i o n a l l y e f f i c i e n t marke ts , " presented a t the Winter Meetings o f the Econometr ic S o c i e t y , D a l l a s . Mimeo. [1976] . " In fo rmat ion and compe t i t i ve p r i c e sys tems , " American Economic Review, 66 #2, 246-253. Hanoch, G. and H. Levy [1969] . "The e f f i c i e n c y a n a l y s i s o f cho ices i n v o l v i n g r i s k , " Review o f Economic S t u d i e s , 36, 335-346. H e i n k e l , Robert [1978] . "Moral haza rd , r i s k y debt and c r e d i t r a t i o n -i n g , " Graduate School o f Bus iness A d m i n i s t r a t i o n , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , January . H i r s c h , M. W. [1976] . D i f f e r e n t i a l Topology. New York : S p r i n g e r -V e r l a g . H i r s h l e i f e r , J . [1971] . "The p r i v a t e and s o c i a l va lue of i n fo rma t ion and the reward to i n v e n t i v e a c t i v i t y , " American Economic Review, 61_, 561-574. J a f f e , Dwight M. and Thomas R u s s e l l [1976] . " Imper fect i n f o r m a t i o n , u n c e r t a i n t y and c r e d i t r a t i o n i n g , " Qua r te r l y Journa l o f Economics, XC #4, 651-665. Jo rdan , J . and R. Radner [1977] . "Example o f nonex is tence of r a t i o n a l expec ta t i ons e q u i l i b r i u m . " Mimeo d r a f t f o r Workshop i n Ra t iona l Expec ta t ions and D i f f e r e n t i a l In format ion IMSSS. S t a n f o r d . J u l y . K raus , A . [1976] . D i s c u s s i o n on Grossman [1976] , Journa l o f F i n a n c e , XXXI #2, 602-604. K raus , A lan and Gordon S i ck [1979a] , " P r i c e s ' as s i g n a l s o f p re ferences and endowments," Working Paper 664, F a c u l t y o f Commerce and Business A d m i n i s t r a t i o n , U n i v e r s i t y o f B r i t i s h Columbia, Vancouver. June. / [ 1979b ] . "Communication o f aggregate pre ferences through market p r i c e s , " Journa l o f F i n a n c i a l and Q u a n t i t a t i v e A n a l y s i s , XIV #4, 699-703. [1980] . " D i s t i n g u i s h i n g b e l i e f s and pre ferences i n e q u i l i b r i u m p r i c e s , " Journa l o f F i n a n c e , 35 #2, 335-344. Kreps , D. M. [1977] . "A note on ' f u l f i l l e d e x p e c t a t i o n s ' e q u i l i b r i a , " Journa l o f Economic Theory, 14, 32-43. 131. L e l a n d , Hayne E. and David H. Py le [1977] , " In fo rmat iona l asymmetr ies, f i n a n c i a l s t r u c t u r e s and f i n a n c i a l i n t e r m e d i a r i e s , " Journa l o f  F inance XXXI I , 371-388. Le Roy, Stephen and Richard Po r t e r [1979] . "The present va lue r e l a t i o n : Tests based on imp l i ed va r iance bounds, " Mimeo., Board of Governors o f the Federal Reserve System. Washington. M a r s h a l l , John [1974] . " P r i v a t e i n c e n t i v e s and p u b l i c i n f o r m a t i o n , " American Economic Review, 64 , 373-390. Mer ton , R. C. [1971] . "Optimum consumption and p o r t f o l i o r u l e s i n a cont inuous t ime mode l , " Journa l o f Economic Theory, 3 #4, 373-413. [1973] . "An in te r tempora l c a p i t a l a s s e t p r i c i n g mode l , " Economet r ica , 41 #5, 867-887. M o s s i n , J . [1968] . "Optimal m u l t i p e r i o d p o r t f o l i o p o l i c i e s , " Journa l o f  B u s i n e s s , 4 1 . 215-229. Muth, John F. [1961] . "Ra t i ona l expec ta t i ons and the theory o f p r i c e movements," Economet r ica , 29 , 315-335. Naga tan i , Keizo [1975] . "On a theorem o f A r row , " Review o f Economic  S t u d i e s , XLII #131, 483-485. Radner, R. [1977] . "Ra t i ona l expec ta t i ons e q u i l i b r i u m : Gener ic e x i s t e n c e and the i n fo rma t ion revea led by p r i c e s , " Center f o r Research i n Management S c i e n c e , U. o f C. B e r k e l e y , W.P. IP -245 . To appear i n Econometr ica . Ross , Stephen A . [1977] . "The de te rmina t ion o f f i n a n c i a l s t r u c t u r e : The i n c e n t i v e s i g n a l l i n g app roach , " B e l l Journa l o f Economics, S p r i n g . [1978a] . "Some notes on f i n a n c i a l i n c e n t i v e - s i g n a l l i n g models, a c t i v i t y cho ice and r i s k p r e f e r e n c e s , " Journa l o f F i nance , XXXIII #3, 777-792. [1978b] . "Mutual fund sepa ra t i on i n f i n a n c i a l theory - - the sepa ra t i ng d i s t r i b u t i o n s , " Journa l o f Economic Theory, 17, 254-286. R o t h s c h i l d , Michael [1976] . "Recent work on e q u i l i b r i u m and expec ta t i ons under u n c e r t a i n t y , " i n "Systemes dynamiques e t modeles economiques," Co l logues I n t e r n a t i o n a l du C . N . R . S . #259, 99-109. R o t h s c h i l d , Michael and Joseph S t i g l i t z [1976] . " E q u i l i b r i u m i n compe t i t i ve insurance markets : An essay on the economics o f imper fec t i n f o r m a t i o n , " Qua r te r l y Journa l o f Economics, XC #4, .629-649. 132. R u b i n s t e i n , M. [1973] . "A comparat ive s t a t i c s a n a l y s i s o f r i s k premiums," Journa l o f B u s i n e s s , 46 , 605-615. [1974] . "An aggregat ion theorem f o r s e c u r i t i e s ma rke t s , " Journa l o f F i n a n c i a l Economics, 1 , 225.-244. Rud in , Wal ter [1964] . P r i n c i p l e s o f Mathematical A n a l y s i s . McGraw H i l l , New York . S a l o p , J . and S . Salop [1976] . " S e l f - s e l e c t i o n and turnover i n the l abo r marke t , " Qua r te r l y Journa l o f Economics, XC #4, 619-627. S h i l l e r , Robert [1978] . "Ra t i ona l expec ta t i ons and the dynamic s t r u c t u r e o f macroeconomic models: A c r i t i c a l r e v i e w , " Journa l o f Monetary  Economics, 4 , #1, January . [1979] , "Do s tock p r i c e s move too much to be j u s t i f i e d by subsequent changes i n d i v i d e n d s ? , " Mimeo. , U n i v e r s i t y o f P e n n s y l v a n i a , Dept. o f Economics. October . Spence, A . M. [1976] . "Job market s i g n a l l i n g , " Qua r te r l y Journa l  o f Economics, LXXXVII #3, 355-374. T a y l o r , S . J . [1973] . I n t roduc t i on to Measure and I n t e g r a t i o n . Cambridge U n i v e r s i t y P r e s s , Cambridge. Van Home, James C. [1978] . F i n a n c i a l Market Rates and F lows. P r e n t i c e - H a l l , Englewood C l i f f s . Z a n g w i l l , W. [1969] . Non l inear Programming - - a U n i f i e d Approach. P r e n t i c e - H a l l , Englewood C l i f f s . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0095184/manifest

Comment

Related Items