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Two essays on financial economics : I. Weighted utility, risk aversion and portfolio choice : II. Competitive.. 1985
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Title | Two essays on financial economics : I. Weighted utility, risk aversion and portfolio choice : II. Competitive bidding and interest rate formation in an informal financial market |
Creator |
Mao, Mei Hui Jennifer |
Publisher | University of British Columbia |
Date Created | 2010-06-23T19:55:16Z |
Date Issued | 2010-06-23T19:55:16Z |
Date | 1985 |
Description | This thesis consists of two essays. Each essay addresses a research problem involving some aspects of uncertainty and financial economics. Essay 1 deals with the general question of whether classical results in risk aversion and portfolio choice based on expected utility hypothesis are robust with respect to recent works in nonlinear utility theories generalizing expected utility. We investigate the implications of an axiomatic generalization called weighted utility theory along with the weaker, but unaxiomatized linear Gateaux utility. We establish the equivalence among three definitions of individual global risk aversion, i.e., in terms of conditional certainty equivalent, mean preserving spread, and conditional risky-asset demand, without any differentiability assumptions about the preference functional. The only requirement is that the preference ordering be complete, transitive, consistent with first-degree stochastic dominance, and continuous in distribution. The equivalence between the first two definitions is also extended to a comparative context. We also identify the necessary and sufficient condition for the single risky asset to be a normal good to a weighted utility maximizer with concave lottery-specific utility functions. Unlike its expected utility counterpart, which depends only on the agent's initial wealth and preferences, this condition also depends on the characteristics of the risky asset. The second essay examines the role of a sequential competitive bidding process in the endogenous determination of interest rates and the corresponding allocation of loans and savings in a widely observed class of informal financial markets called the 'rotating credit association'. Optimal bidding strategies are obtained for individual agents with concave and time-additive utility functions. After deriving some comparative statics and efficiency implications of the individual optimal bidding strategy, we impose further restrictions, including risk neutrality, to obtain a tractable form of a Nash equilibrium bidding strategy. This yields, for each agent, an ex post borrowing, as well as lending, interest rate depending on the history of the realized winning bids, including the one for the period in which he won the auction. Weighted by the Nash equilibrium-induced probability of winning in each period, ex ante borrowing and lending interest rates result. |
Subject |
Risk Rotating Credit Associations |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2010-06-23T19:55:16Z |
DOI | 10.14288/1.0096730 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/25936 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0096730/source |
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TWO ESSAYS ON FINANCIAL ECONOMICS: , WEIGHTED UTILITY, RISK AVERSION AND PORTFOLIO CHOICE II . COMPETITIVE BIDDING AND INTEREST RATE FORMATION IN AN INFORMAL FINANCIAL MARKET by MAO, MEI HUI JENNIFER B.Comm., The National Taiwan Uni v e r s i t y , 1975 M.B.A., The National Cheng-Chi Uni v e r s i t y , 1978 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Faculty of Commerce and Business Administration) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1985 © Mao, Mei Hui Jennifer In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Commerce & Business Administra t i o n The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date July 16, 1985 /«i ^ Supervisor: Dr. Robert A. Jones Abstract This thes is cons is ts of two essays. Each essay addresses a research problem invo lv ing some aspects of uncerta inty and f i n a n c i a l economics. Essay I deals with the general question of whether c l a s s i c a l r e su l t s i n r i s k aversion and p o r t f o l i o choice based on expected u t i l i t y hypothesis are robust with respect to recent works i n nonl inear u t i l i t y theories g e n e r a l i z i n g expected u t i l i t y . We inves t igate the impl i ca t ions of an axiomatic g e n e r a l i z a t i o n c a l l e d weighted u t i l i t y theory along with the weaker, but unaxiomatized l i n e a r Gateaux u t i l i t y . We e s t a b l i s h the equivalence among three d e f i n i t i o n s of i n d i v i d u a l g l o b a l r i s k avers ion , i . e . , i n terms of c o n d i t i o n a l c e r t a i n t y equivalent , mean preserving spread, and c o n d i t i o n a l r i sky -as se t demand, without any d i f f e r e n t i a b i l i t y assumptions about the preference f u n c t i o n a l . The only requirement i s that the preference ordering be complete, t r a n s i t i v e , cons is tent with f i r s t - d e g r e e s tochast ic dominance, and continuous i n d i s t r i b u t i o n . The equivalence between the f i r s t two d e f i n i t i o n s is also extended to a comparative context . We also i d e n t i f y the necessary and s u f f i c i e n t condi t ion for the s ing l e r i s k y asset to be a normal good to a weighted u t i l i t y maximizer with concave l o t t e r y - s p e c i f i c u t i l i t y funct ions . Unl ike i t s expected u t i l i t y counterpart , which depends only on the agent's i n i t i a l wealth and i i preferences , th i s condi t ion a lso depends on the c h a r a c t e r i s t i c s of the r i s k y asset . The second essay examines the ro l e of a sequent ia l competit ive b i d - ding process i n the endogenous determination of in teres t rates and the corresponding a l l o c a t i o n of loans and savings i n a widely observed c lass of informal f i n a n c i a l markets c a l l e d the ' r o t a t i n g c r e d i t a s s o c i a t i o n ' . Optimal bidding s trateg ies are obtained for i n d i v i d u a l agents with concave and t ime-addi t ive u t i l i t y funct ions . A f t e r d e r i v i n g some comparative s t a t i c s and e f f i c i e n c y impl ica t ions of the i n d i v i d u a l optimal b idding s trategy , we impose further r e s t r i c - t i o n s , inc lud ing r i s k n e u t r a l i t y , to obtain a t rac tab le form of a Nash e q u i l i b r i u m bidding s trategy . This y i e l d s , for each agent, an ex post borrowing, as wel l as l ending , in t ere s t rate depending on the h i s t o r y of the r e a l i z e d winning b ids , inc lud ing the one for the period i n which he won the auc t ion . Weighted by the Nash equi l ibr ium- induced p r o b a b i l i t y of winning i n each p e r i o d , ex ante borrowing and lending in teres t rates r e s u l t . H i TABLE OF CONTENTS ABSTRACT i i ESSAY I WEIGHTED UTILITY, RISK AVERSION AND PORTFOLIO CHOICE 1 0 INTRODUCTION 2 0.1 Expected U t i l i t y and Finance: H i s t o r y 2 0.2 A l t e r n a t i v e Preference Theories 6 0.3 Organizat ion of the Essay 10 1 PREFERENCE REPRESENTATION AND STOCHASTIC DOMINANCE 14 1.1 Linear Gateaux U t i l i t y , L inear Impl i c i t U t i l i t y and Weighted U t i l i t y 15 1.2 Frechet D i f f e r e n t i a b l e U t i l i t y 18 1.3 Representation 20 1.4 Stochast ic Dominance 27 2 INDIVIDUAL RISK AVERSION 33 2.1 Local Risk Avers ion 34 2.2 Global Risk Aversion 40 3 PORTFOLIO CHOICE PROBLEM 58 4 COMPARATIVE RISK AVERSION 67 4.1 D e f i n i t i o n s 67 4.2 Character izat ions 72 5 DECREASING RISK AVERSION AND THE NORMALITY OF RISKY-ASSET DEMAND WITH DETERMINISTIC WEALTH 87 5.1 Expected U t i l i t y 87 5.2 Non-Expected U t i l i t y 89 6 COMPARATIVE AND DECREASING RISK AVERSION INVOLVING STOCHASTIC WEALTH 103 6.1 Expected U t i l i t y 105 6.2 Beyond Expected U t i l i t y 110 7 CONCLUSION 117 REFERENCES 124 iv ESSAY II COMPETITIVE BIDDING AND INTEREST RATE FORMATION IN AN INFORMAL FINANCIAL MARKET 130 0 INTRODUCTION 131 1 THE GENERAL STRUCTURE AND ACTUAL CASES OF HUI 138 1.1 The General Structure of Hui 138 1.2 Ac tua l Cases of Hui 141 2 THE ECONOMICS OF HUI WITH TWO OR THREE MEMBERS 157 Two-Member Hui without an Organizer 157 Two-Member Hui with an Organizer 161 Three-Member Hui without an Organizer 162 3 THE MODEL FOR AN N-MEMBER HUI WITH AN ORGANIZER 166 Notations 166 Assumptions 166 4 OPTIMAL INDIVIDUAL BIDDING STRATEGIES 171 4.1 F i r s t - P r i c e Competitive Bidding 172 4.2 Second-Price Competitive Bidding 175 4.3 Impl icat ions 177 5 A NASH PROCESS OF INTEREST RATE FORMATION 183 6 AN APPLICATION TO COLLUSION AMONG SEVERAL SELLERS UNDER REPEATED AUCTIONS 192 6.1 The Structure of Rotat ing Credi t C o l l u s i o n 192 6.2 Assumptions 193 6.3 Nash E q u i l i b r i u m Bidding Strategies 195 7 CONCLUSION 198 REFERENCES 202 APPENDIX 205 Proof of Lemma 3.1 205 Proof of Theorem 4.1 206 Proof of C o r o l l a r y 4.5 208 Proof of C o r o l l a r y 4.7 208 Proof of Theorem 5.1 209 v LIST OF TABLES Table 0.1: Examples of Rota t iona l Exchange 133 Table 1.1: Cash Flow Patterns of Hui P a r t i c i p a n t s 140 Table 1.2: Ac tua l Cases of Hui 143 Table 2.1: Ex Post Interes t Rates and P r o f i t s for A 2-member Hui - An Example 160 Table 3.1: P a r t i c i p a n t i ' s Ind i f f erent Cash Flow Patterns i n a Discount -Bid Hui 168 Table 5.1: Nash Bidding Strategies and Their Der iva t ives 188 v i ACKNOWLEDGMENTS I wish to thank my supervisory committee - Professors Robert Jones, Neal Stoughton, and John Weymark - as wel l as Professors A. Atnershi, Mukesh Eswaran and Hugh Neary for t h e i r h e l p f u l comments and suggestions. I am e s p e c i a l l y gra te fu l to my supervisor Dr. Robert Jones and Professor John Weymark for t h e i r support and encouragement. I have also benefited from valuable discuss ions with Dr. Chew Soo Hong. The s e c r e t a r i a l ass istance of Miss Col leen Colclough is deeply appreciated. E S S A Y I W E I G H T E D U T I L I T Y , R I S K A V E R S I O N A N D P O R T F O L I O C H O I C E 0 INTRODUCTION 0.1 Expected U t i l i t y and Finance: H i s t o r y Given the nature of the topics i t s tudies , finance as a d i s c i p l i n e needs t r a c t a b l e , and yet r i c h enough theories about preferences under un- c e r t a i n t y . Due to i t s s i m p l i c i t y , expected value was once popular as a c r i t e r i o n for dec i s ion making under uncer ta in ty . Investors ' r i s k aver- s ions , evident in the purchase of various types of insurance, the d i v e r s i - f i c a t i o n of p o r t f o l i o s , e t c . , however cast doubts on i t s t h e o r e t i c a l and behav iora l v a l i d i t y . In response, two approaches, namely the mean-vari - ance analys is and the expected u t i l i t y theory, emerged as improved c r i t e - r i a for dec i s ion making under u n c e r t a i n t y . F i r s t invest igated by Tetens (1789), mean-variance analys is had i t s impact on finance only after the works of Markowitz (1952a, 1959) and Tobin (1958), and has since l a i d the foundation for modern p o r t f o l i o theory. Under c e r t a i n assumptions, Sharpe (1964), L in tner (1965) and Treynor (1961) derived the c a p i t a l asset p r i c i n g model which es tabl i shes the l i n e a r r e l a t i o n s h i p between the expected rate of return on a r i s k y asset and i t s market r i s k . The s i m p l i c i t y and i n t u i t i v e n e s s of mean- variance analys is has led to i t s widespread acceptance in f inance . P a r a l l e l and complementary to mean-variance analys is i s the develop- ment of expected u t i l i t y which was f i r s t axiomatized by Ramsey (1926), rev ived by von Neumann and Morgenstern (1947), and ref ined by Marshak - 2 - (1950), Samuelson (1952), Hers te in and Milnor (1953), Savage (1954), Anscombe and Aumann (1963), P r a t t , R a i f f a and S c h l a i f e r (1964), DeGroot (1970), Fishburn (1970), Arrow (1971) and others . Derived from a set of p l a u s i b l e axioms that lead to a simple, t rac tab le representat ion form, ex- pected u t i l i t y has provided the foundation for the microeconomics of un- c e r t a i n t y in the las t three decades. Despite i t s t r a c t a b i l i t y , mean-variance analys is suffers from several t h e o r e t i c a l weaknesses. S p e c i f i c a l l y , Borch (1969) and F e l d s t e i n (1969) showed that mean-variance analys is v i o l a t e s s tochast ic dominance ( i . e . , a l o t t e r y may be preferred to another l o t t e r y that always d e l i v e r s a better outcome with a higher p r o b a b i l i t y ) i n the sense that , given an i n d i f f e r - ence curve, one can always construct in the be t ter - than region a l o t t e r y which is s t o c h a s t i c a l l y dominated by a l o t t e r y on the given ind i f f erence curve. It i s also wel l known that , under expected u t i l i t y theory, mean- variance analys is is v a l i d only i f one of the fol lowing two assumptions holds: (1) agents have quadratic von Neumann-Morgenstern u t i l i t y func- t i o n s ; (2) the underly ing random v a r i a b l e i s normally d i s t r i b u t e d . E i t h e r Borch (1969) showed that , given two l o t t e r i e s A and B i n d i f f e r e n t in the mean-variance sense, we can always construct another pa ir of l o t t e r i e s A ' and B 1 such that u.̂ = |J^i , o"A = a^, , Mg = Mgi , Og = Ogf , but A' s to- c h a s t i c a l l y dominates B' . S p e c i f i c a l l y , l e t u.̂ > Lig, ô > Og, and p = ( ( V ^ ) 2 / [ ( ^ A - ^ ) 2 + ( a A - ° B ) 2 ] > X = ^ A - ^ ^ V V ' x = > V H = XXaB> ^ 1 = ^ A + °A^ X> Y2 = ^ 1 B + C J B / ^ - 1^ A' i s t n e l o t t e r y of ge t t ing x with 1-p chance and g e t t i n g y^ with p chance and B' is the l o t t e r y of get t ing x w i t h 1-p chance and g e t t i n g y2 with p chance, then A' and B' are such two l o t t e r i e s . - 3 - requirement is not s a t i s f a c t o r y . The normal d i s t r i b u t i o n i s at times overly r e s t r i c t i v e in modelling finance phenomena. The quadratic u t i l i t y function, on the other hand, i s unappealing because i t requires a bounded domain and implies increasing absolute r i s k aversion. In the area of finance, the following are l i n e s of research that have d i r e c t t i e s with expected u t i l i t y theory. Pratt (1964) and Arrow (1971) started the l i t e r a t u r e now known as the theory of r i s k aversion. Pratt characterized r i s k aversion by r i s k premium and, for an expected u t i l i t y d ecision maker with von Neumann-Morgenstern u t i l i t y function u, i d e n t i f i e d indexes -u"(x)/u'(x) and -xu"(x)/u'(x) as the measure of absolute and re- l a t i v e l o c a l r i s k aversion, r e s p e c t i v e l y (the former i s now known as the Arrow-Pratt index). He further characterized comparative r i s k aversion v i a r i s k premium, p r o b a b i l i t y premium, and the Arrow-Pratt index. He also i d e n t i f i e d the class of u t i l i t y functions which w i l l exhibit decreasing, constant, or increasing absolute/relative r i s k aversion, as well as the operations that w i l l preserve such properties. Arrow independently established j u s t i f i c a t i o n s for decreasing abso- lute r i s k averison and increasing r e l a t i v e r i s k aversion. He also charac- t e r i z e d , in terms of decreasing absolute r i s k aversion, the normality of risky-asset demand in a one-safe-asset-one-risky-asset world where i n i t i a l wealth i s deterministic. In addition, the safe asset w i l l be a luxury good i f the investor's r e l a t i v e r i s k aversion i s increasing in his wealth. Cass and S t i g l i t z (1970; 1972) extended Arrow's in v e s t i g a t i o n of wealth e f f e c t s into the world of many ris k y assets. They showed that, i f the agent's preference exhibits a 'separation property' so that the r i s k y - 4 - assets as a whole can be viewed as a mutual fund, then Arrow's normality r e s u l t w i l l continue to hold for these r i s k y assets c o l l e c t i v e l y . They also i d e n t i f i e d the c lass of u t i l i t y functions that have the separation property to e i ther be quadratic or d i sp lay constant r e l a t i v e r i s k aver- s i o n . Hart (1975) further proved that the separat ion property is both necessary and s u f f i c i e n t for mul t ip l e r i s k y assets to be normal. A l t e r n a t i v e l y , by r e s t r i c t i n g u t i l i t y functions to those represent ing the same o r d i n a l preferences , Kih l s trom and Mirman (1974) were able to ge- n e r a l i z e Arrow and P r a t t ' s c h a r a c t e r i z a t i o n of comparative r i s k aversion to a world of many commodities and thereby inves t igate i t s impl icat ions for agents' consumption and savings choice . Motivated by Kih l s trom and Mirman's d e f i n i t i o n of r i s k avers ion , Paroush (1975) further proposed a n a t u r a l genera l i za t ion of Arrow-Prat t ' s r i s k premium to the world of mul- t i p l e commodities. In another d i r e c t i o n , Levy and K r o l l (1978), Ross (1981), and K i h l s t r o m , Romer and Will iams (1981) invest igated the case where the i n - v e s t i b l e i n i t i a l wealth is random rather than d e t e r m i n i s t i c . It was shown that , un l ike the case of de termin i s t i c i n i t i a l wealth, a more r i s k averse ( i n the sense of Arrow-Pratt ) i n d i v i d u a l need not be w i l l i n g to pay a higher premium to insure against a given r i s k than his less r i s k averse counterpart . Ross (1981) proposed a stronger condi t ion under which Arrow- Prat t 's c h a r a c t e r i z a t i o n of comparative r i s k aversion w i l l carry through even i f the i n i t i a l wealth is random. While Arrow and Prat t focused t h e i r a t tent ion on the r e l a t i o n between propert ies of the u t i l i t y funct ion and the behavioral impl icat ions of r i s k avers ion , Rothschi ld and S t i g l i t z (1970; 1971) character ized a dec i s ion - 5 - maker's r i s k aversion by his response to a p a r t i c u l a r type of increase in r i s k , termed 'mean perserving spread' . They showed that the concavity of u t i l i t y functions and the preference for a d i s t r i b u t i o n over i t s mean pre- serving spreads are equivalent in t h e i r c h a r a c t e r i z a t i o n of r i s k avers ion . Moreover, each of these is more general than the once popular variance c r i t e r i o n in the sense that the former i m p l i e s , but i s not implied by, the l a t t e r . Fol lowing Rothschi ld and S t i g l i t z ' mean preserving spread con- cept , Diamond and S t i g l i t z (1974) proposed a s i m i l a r not ion c a l l e d 'mean u t i l i t y preserving spread' which is useful for c h a r a c t e r i z i n g comparative r i s k aversion for preferences that are l i n e a r in p r o b a b i l i t y d i s t r i b u t i o n . The above inves t igat ions along with others not mentioned here c o n s t i - tute the l i t e r a t u r e on r i s k aversion which r e l i e s c r i t i c a l l y on the assum- pt ion that the agent i n question maximizes h is expected u t i l i t y . 0.2 A l t e r n a t i v e Preference Theories While expected u t i l i t y seems to have served finance w e l l , i t s e m p i r i - c a l v a l i d i t y has been questioned by dec i s ion s c i e n t i s t s in l i g h t of c e r - t a i n widely reported v i o l a t i o n s of i t s i m p l i c a t i o n s , inc lud ing the A l l a i s paradox and the concurrence of r i s k - s e e k i n g and r i s k - a v e r t i n g behavior wi - t h i n an i n t e r v a l of monetary outcomes. Given the prevalence and p e r s i s t - ence of these phenomena, the quiet acceptance of expected u t i l i t y in f i - nance should not be in terpre ted as unreserved content. Rather, as w i l l be i l l u s t r a t e d below, i t i s mainly due to i t s t r a c t a b i l i t y and the lack of a v i a b l e a l t e r n a t i v e preference theory. We w i l l , i n th is s ec t ion , touch upon several a l t e r n a t i v e theories that have been reported in the l i t e r a - - 6 - ture and then, in the remainder of th is essay, focus on two approaches which hold promise to open some new paths for i n v e s t i g a t i n g f i n a n c i a l eco- nomics beyond expected u t i l i t y . M i s p e r c e p t i o n - o f - P r o b a b i l i t y Theories This l i n e of i n v e s t i g a t i o n s , s tarted by Edwards (1954), attempts to account for the reported expected u t i l i t y anomalies by rep lac ing the pro- b a b i l i t y weights p^ in the expected u t i l i t y expression for a f i n i t e l o t t e - r y , i . e . Ep^u(x^) , by a n o n l i n e a r f u n c t i o n f(p^) (often in terpreted as some subject ive p r o b a b i l i t i e s ) which may not add up to u n i t y . Subsequent adherents to th is view include Handa (1977), who adopted the form Ef(p^)x^, Karmarkar (1978), who normalized the weights using the form E[ f (p^ ) / E f ( p̂ , ) ] u(x^ ) , and Kahneman and Tversky (1979), who edited l o t t e r i e s before dec iding which of t h e i r two evaluat ion equations to be a p p l i e d . There are at least three problems with any approach whose represen- t a t i o n takes the form £ f ( p ^ , x ^ ) . F i r s t of a l l , i t can only apply to f i - n i t e l o t t e r i e s . To see t h i s , suppose the p r o b a b i l i t y d i s t r i b u t i o n is con- t i n u o u s over [ a , b ] . I t i s not known how we can ca l cu la te Ef(p^,x^) or / f ( p ^ , x ^ ) . 2 Since finance is frequently concerned with problems invo lv ing continuous d i s t r i b u t i o n s , th is i s a serious l i m i t a t i o n that casts doubt on t h e i r appeal to f i n a n c i a l economists as a general dec i s ion r u l e . This i s because the l i m i t of the value E f ( p £ , x ^ ) or / f ( p ^ , X £ ) of any se- quence of f i n i t e l o t t e r i e s converging to a continuous p r o b a b i l i t y d i s - t r i b u t i o n does not e x i s t . - 7 - Another problem with any Edwardsian theory is i t s inherent tendency to v i o l a t e s tochast ic dominance. Kahneman and Tversky (1979) added the detection-of-dominance operation to circumvent th i s d i f f i c u l t y but paid a steep pr ice in v i o l a t i n g t r a n s i t i v i t y (Chew, 1980; Machina, 1982a). The t h i r d problem with the m i s p e r c e p t i o n - o f - p r o b a b i l i t y approach l i e s i n i t s i n a b i l i t y to d i sp lay g lobal r i s k aversion except when f(p^) = p^, i n which case the expected u t i l i t y obta ins . Global r i s k avers ion , in the sense of preference for a d i s t r i b u t i o n over any of i t s mean-preserving- spreads ( i . e . d i s t r i b u t i o n s with equal mean but higher v a r i a b i l i t y ) is under expected u t i l i t y equivalent to pointwise l o c a l r i s k aversion ( i . e . avers ion towards a c t u a r i a l l y f a i r i n f i n i t e s i m a l r i s k s ) . Since g lobal r i s k avers ion i s regarded as an appealing property in f inance , f i n a n c i a l econo- mists might be re luc tant to replace expected u t i l i t y by theories which are inherent ly unable to d i sp lay such a preference property . General Preference Funct iona l s Motivated by the paradoxes i n his name, A l l a i s i s probably the f i r s t to consider a preference funct iona l more general than that of expected u t i l i t y . He argued that a dec i s i on maker's preference is represented by a funct ion of the moments of the d i s t r i b u t i o n of some ' c a r d i n a l psycho log i - c a l value' funct ions . Suppose V is the A l l a i s ' preference f u n c t i o n a l . Then V = V ( m ^ , m 2 , . . . ) , where m. = Js ( x ) 1 d F ( x ) . This model i s indeed very general and contains both expected u t i l i t y - 8 - and mean-variance analys is as spec ia l cases. When the preference depends o n l y on the f i r s t moment m̂ , expected u t i l i t y obta ins . In A l l a i s ' view, expected u t i l i t y ' s i n a b i l i t y to describe A l l a i s - t y p e choice behavior is because higher moments are unduly ignored. Another spec ia l case — mean- var iance analys is — resu l t s when the c a r d i n a l psycholog ica l value func- t i o n s(x) i s l i n e a r in the underlying monetary outcome x, and only the f i r s t two moments matter. It is i n t e r e s t i n g to note that , i f s(x) is non l inear , then V = VCm^jii^) becomes the preference funct iona l for a ' u t i l i t y mean-variance' analys i s — a case yet to be explored. One poss ib le d i r e c t i o n is in deve- loping a ' u t i l i t y - m e a n - v a r i a n c e ' c a p i t a l asset p r i c i n g model i n l i g h t of the success and tenac i ty of the present mean-variance-based c a p i t a l asset p r i c i n g model. Based on Borch (1969)'s i l l u s t r a t i o n , however, i t is c l ear that u t i l i t y - m e a n - v a r i a n c e a n a l y s i s , l i k e mean-variance a n a l y s i s , w i l l also v i o l a t e s tochast ic dominance. Two other a l t e r n a t i v e theories to expected u t i l i t y also belong in the category of general preference funct ionals - one is Machina's (1982a) 'ex- pected u t i l i t y a n a l y s i s ' without the independence axiom, the other is weighted u t i l i t y theory recent ly proposed in Chew and MacCrimmon (1979a & b ) , Chew (1980; 1981; 1982; 1983), Fishburn (1983) and Nakamura (1984). The rest of the essay w i l l be devoted to de ta i l ed discuss ions on t h e i r as- sumptions, representat ion forms, as well as resu l t s p o t e n t i a l l y i n t e r e s t - ing to f i n a n c e . J a Other approaches not considered here include the information processing models ( e .g . Payne, 1973), Meginniss' (1977) 'entropy' preference, and the theory of regret proposed by B e l l (1982) and Loomes and Sugden (1983). - 9 - 0.3 Organizat ion of the Essay This essay focuses on comparisons of three preference theor ie s , name- l y expected u t i l i t y , weighted u t i l i t y and l i n e a r Gateaux u t i l i t y , i n terms of t h e i r condi t ions , propert ies and impl ica t ions of r i s k avers ion . In Sect ion 1, we summarize t h e i r representat ion func t iona l forms. To avoid mathematic t e c h n i c a l i t i e s , they are a l l stated as hypotheses. We regard the consistency with f i r s t - d e g r e e s tochast ic dominance as a property that any usefu l preference theory must possess. The condi t ion for th i s proper- ty under d i f f e r e n t theories are g iven . In Sect ion 2 , we inves t iga te the propert ies and impl i ca t ions of i n d i - v i d u a l r i s k avers ion , i n c l u d i n g l o c a l r i s k avers ion and g loba l r i s k aver- s i o n . The d i s t i n c t i o n i s important because they do not co inc ide beyond expected u t i l i t y . D i f f e r e n t notions of r i s k aversion are def ined . Among them are r i s k avers ion i n the sense of c o n d i t i o n a l and uncondi t ional c e r - t a i n t y equivalent , r i s k aversion i n terms of mean preserving spread, and pointwise l o c a l r i s k avers ion . Risk aversion i n terms of c o n d i t i o n a l c e r - t a i n t y equivalent and r i s k avers ion i n terms of mean preserving spread are shown to be equivalent regardless of the underlying preference theory. This r e s u l t only requires the preference to be complete, t r a n s i t i v e , con- s i s t e n t with f i r s t - d e g r e e s tochast ic dominance and continuous i n d i s t r i b u - t i o n . On the contrary , some of the r i s k aversion d e f i n i t i o n s are only equivalent under expected u t i l i t y . Sect ion 3 introduces p o r t f o l i o choice problem i n a world with one safe asset and one r i s k y asset . I t i s showed that a r i s k averse agent w i l l never short s e l l the r i s k y asset as long as i t s expected re turn i s - 10 - s t r i c t l y greater than the safe return. When they are equal, his best strategy i s to invest only in the safe asset. If he i s a weighted u t i l i t y maximizer, then he w i l l invest a p o s i t i v e amount in the single r i s k y asset i f and only i f i t s expected return i s s t r i c t l y greater than the safe return. We further show that, i f an agent w i l l invest in the r i s k y asset only i f i t s expected return is s t r i c t l y greater than the safe return, then he must be g l o b a l l y r i s k averse i n the sense of conditional certainty equiva- l e n t . This amounts to the equivalence between global c o n d i t i o n a l - c e r t a i n - ty-equivalent r i s k aversion and global conditional p o r t f o l i o r i s k aversion for any preference ordering s a t i s f y i n g completeness, t r a n s i t i v i t y and f i r s t - d e g r e e stochastic dominance. Section 4 characterizes comparative r i s k aversion across i n d i v i d u a l s . Again, we prove that comparative r i s k aversion in terms of conditional c e r t a i n t y equivalent is equivalent to comparative r i s k aversion i n terms of mean preserving spread without depending on s p e c i f i c u t i l i t y functional forms. In Section 5, we .derive the necessary and s u f f i c i e n t condition for the risky asset to be a normal good for a weighted u t i l i t y maximizer. This r e s u l t u t i l i z e s the e x p l i c i t functional form of weighted u t i l i t y , and i s not obtainable under l i n e a r Gateaux u t i l i t y . In Section 6, stochastic wealth is introduced to the characterization of decreasing and comparative r i s k aversion. It appears that weighted u t i l i t y and l i n e a r Gateaux u t i l i t y provide more room in allowing addition- a l r i s k s because t h e i r u t i l i t y functions are l o t t e r y - s p e c i f i c . The expec- ted u t i l i t y r esults in this section are from Ross (1981). The r e s u l t s be- yond expected u t i l i t y were f i r s t proved by Machina (1982b) for Frechet d i - ff e r e n t i a b l e u t i l i t y and l a t e r extended by Chew (1985) to l i n e a r Gateaux u t i l i t y . They are reproduced mainly to complete the spectrum of our com- parisons . In this essay, d e f i n i t i o n s , expressions, lemmas and c o r o l l a r i e s are numbered according to the section and the order i n which they appear. For the convenience of comparisons across d i f f e r e n t preference theories, theo- rems are in general l a b e l l e d with U, EU, WU or LGU, followed by an Arabic number. Obviously, EU, WU and LGU stand for expected u t i l i t y , weighted u t i l i t y , l i n e a r Gateaux u t i l i t y , r e s p e c t i v e l y . U is used when the result does not depend on a s p e c i f i c preference functional form. The Arabic num- ber indicates the nature of the r e s u l t . A summary is given below: Theorem No. Regarding 1 representation 2 first-degree stochastic dominance (SD) 3 Arrow-Pratt index 4 pointwise l o c a l r i s k aversion (PLRA) 5 global r i s k aversion (GRA) 6 nonnegative or p o s i t i v e conditional risky-asset demand 7 comparative r i s k aversion with deterministic wealth (CRA) 8 decreasing r i s k aversion with deterministic wealth (DRA) 9 comparative r i s k aversion with stochastic wealth 10 decreasing r i s k aversion with stochastic i n i t i a l wealth but deterministic wealth increments 11 decreasing r i s k aversion with stochastic i n i t i a l wealth and wealth increments - 12 - When there are more than one theorem on the same subject , decimal f r a c - t ions are used in l a b e l l i n g them. These re su l t s are b r i e f l y summarized in Sect ion 7 which then concludes th is essay by suggesting some p o t e n t i a l app l i ca t ions of non-expected u t i l i t y theories i n the area of f i n a n c i a l economics. - 13 - 1 P R E F E R E N C E R E P R E S E N T A T I O N A N D S T O C H A S T I C D O M I N A N C E To obtain a preference representat ion , one can i d e n t i f y a set of normatively appealing axioms about preferences , then construct a p r e f e r - ence funct iona l which s a t i s f i e s these axioms. A l t e r n a t i v e l y , one can s tar t with a general preference ordering with very l i t t l e a p r i o r i r e s - t r i c t i o n s , and inves t igate sys temat ica l ly the impl icat ions of success ive ly imposed s t r u c t u r e s . While the h i s t o r i c a l development of expected u t i l i t y conformed to the f i r s t approach, the l a t t e r i s useful in providing i n - s ights into the meaning of some preference impl icat ions which are derived from expected u t i l i t y but may not be sens i t ive to the underlying theore- t i c a l s t r u c t u r e s . Let D denote a space of p r o b a b i l i t y measures on some outcome set without pre-imposed r e s t r i c t i o n s . The weakest requirement for a p r e f e r - ence ordering is the customary completeness and t r a n s i t i v i t y . We suppose that such a preference ordering is represented by a u t i l i t y funct iona l V: D + R so that , for any F , G e D, F is weakly preferred to G i f and only i f V(F) > V(G) . This ru les out any l ex i co -graph ic type preferences . A preference representat ion at th i s l eve l of genera l i ty is of l i t t l e i n t e r e s t . To i d e n t i f y des irab le s tructure to impose on V, define F a = ( l - a ) F + OG, where a e [ 0 , 1 ] . ( 1 . 1 ) In o ther words , F t t i s a p r o b a b i l i t y mixture of F and G. As a increases from 0 to 1 . F a goes from F to G. The d e r i v a t i v e ^— F " = G-F is c a l l e d ° da the ' d i r e c t i o n ' of F a . The f i r s t s tructure one would consider to impose - 1 4 - on V i s n a t u r a l l y some sort of smoothness of V(F ) as F s h i f t s from F to G. Assumption 1.1: V(F t t) is d i f f e r e n t i a b l e i n a. So f ar, we have considered p r o b a b i l i t y measures defined on some outcome space which is very general and may be non-numerical. I f we are simply interested in monetary outcomes, we may only consider p r o b a b i l i t y measures defined on some i n t e r v a l J of the re a l l i n e R, allowing J = R as a special case. We denote by D T the space of such d i s t r i b u t i o n s . 1.1 Linear Gateaux U t i l i t y , Linear Implicit U t i l i t y and Weighted U t i l i t y We may fu r t h e r require that 4— V(F t t) take a s p e c i f i c form. Consider n • da r the following: Assumption 1.2: For every F e D T, there exists a function £(•;•): J * D -> R such that, for every F a = (l-a)F+aG, a e [0,1], ^ V ( F a ) = /C(x;F a)d[G(x)-F(x)] . (1.2) In functional analysis, " J ^ V ^ 0 ) i - s c a l l e d the 'Gateaux d i f f e r e n t i a l ' , and C(x;F t t) the 'Gateaux d e r i v a t i v e ' or the ' d i r e c t i o n a l d e r i v a t i v e ' , of V(F t t) at F * i n the d i r e c t i o n of G-F ( L u e n b e r g e r , 1969). For a u t i l i t y functional V s a t i s f y i n g Assumption 1.2, the Gateaux d i f f e r e n t i a l i s l i n e a r i n the d i r e c t i o n G-F and may therefore be c a l l e d a l i n e a r Gateaux preference functional or l i n e a r Gateaux u t i l i t y . As i t turns out, a subclass of l i n e a r Gateaux preference functionals V can be i m p l i c i t l y defined by the following: - 15 - J<t>(x,V(F))dF(x) = 0 , (1.3) where (!>: J2 •*• R i s increas ing in x and decreasing in V ( F ) . This c lass of funct ionals is c a l l e d l i n e a r i m p l i c i t u t i l i t y in Chew ( 1 9 8 4 ) . One such example i s given by the fo l lowing: <Kx,V(F)) = w(x) [v (x ) -V(F) ] ; ( 1 . 4 ) or e q u i v a l e n t l y , <Kx,m) = w(x)[v(x)-v(m)] , (1.5) where m is the c e r t a i n t y equivalent of d i s t r i b u t i o n F . <|)(x,V(F)) is in a sense a ' u t i l i t y dev ia t ion ' of an outcome x from the c e r t a i n t y equivalent of F . Let A ( M(V\\ - o4>(x,V(F)) 4>2(x,V(F)) = — m f ) • It can be v e r i f i e d that - j f f i f f i f e ^ E « * . « « > = J * *R ( 1 . 6 , or ^yALV— = c ( x ; F ) : j x Y> •*• R ( i . 7 ) - /<D 2 (x,v(F))dF(x) - J UJ is the Gateaux d e r i v a t i v e of the l i n e a r i m p l i c i t u t i l i t y V (defined by ( 1 . 3 ) ) at F . This example turns out to be weighted u t i l i t y — a g e n e r a l i z a t i o n of expected u t i l i t y . S p e c i f i c a l l y , weighted u t i l i t y is a subclass of l i n e a r i m p l i c i t u t i l i t y with the preference funct ional V being e x p l i c i t l y given by iT(v\ - mrCr^ - /w( x)v (x)dF(x) 0 v V ( F ) " W J ( F ) " /w(x)dF(x) ' ( l - 8 ) where w(x) is s t r i c t l y p o s i t i v e and c a l l e d a 'weight f u n c t i o n 1 , v(x) i s - 1 6 - s t r i c t l y increasing and c a l l e d a 'value function'. With the s p e c i f i c form of V(F), we can obtain i t s Gateaux d i f f e r e n t i - a l as follows: dV(F a) dWU(Fa) da da lira 4 [ w u(F a +°)-WU(F a) j 9-K) ,. 1 i/vwdF 0^^ /vwdFa-i lim -5 * - 9+0 /wdF /wdF 1 I /vwdF /wdF - /vwdF /wdF i lim L 5 J 9+0 9 /wdF a + 9 /wdFa . 1 r 9/vwd [ G-F ] /wdF - /vwdF 9/wd [ G-F ] lim - 5 [- 9+0 9 /wdF a/wdFa+9/wd [G-F]/wdF a /vwd [ G-F ] /wdF /vwdF g/wd [ G-F ] /wdF "/wdF a /vwd[G-F]-WU(Fa) /wd[G-F] /wdF" /[v(x)w(x)-WU(F a)w(x)]d[G(x)-F(x)] /w(x)dF a(x) /C(x;F a)d[G(x)-F(x)], (1.9) where « x ; F ° ) • w(x)[v(x)-WU(F g)] ^ ( 1 > 1 Q ) /w(x)dF a(x) Thus, we have v e r i f i e d f i r s t of a l l that the Gateaux d i f f e r e n t i a l of a weighted u t i l i t y functional is l i n e a r i n the d i r e c t i o n G-F. Secondly, since /C(x;F)dF(x) = 0, we have shown that weighted u t i l i t y is indeed a special case of lin e a r i m p l i c i t u t i l i t y . - 17 - 1.2 Frechet Dif ferent iable U t i l i t y Instead of Assumption 1.2, Machina (1982a) assumes that , in moving from F to G, V is smooth i n the sense of Frechet d i f f e r e n t i a b i l i t y , i . e . , V(G) - V(F) = Lp(G-F) + ollG-Fll, (1.11) where is some l i n e a r funct iona l which depends on F . This is equivalent to assuming that , corresponding to each l o t t e r y F , there ex is t s a funct ion u(x;F) such that L F ( G - F ) = Ju (x;F)d[G-F] = /u(x;F)dG - Ju (x ;F)dF. (1.12) In other words, V(G)-V(F) can be approximated by the d i f ference in the 'expected u t i l i t i e s ' of G and F using u(x;F) as a ' l o c a l u t i l i t y f u n c t i o n ' . Given (1.11) and (1 .12) , i t can be v e r i f i e d that ^ V ( F a ) | c p 0 = / u ( x ; F ) d [ G ( x ) - F ( x ) ] . (1.13) Thus Machina's Frechet d i f f e r e n t i a b l e u t i l i t y is also a subclass of l i n e a r Gateaux u t i l i t y . It is less general than the l a t t e r i n r e q u i r i n g the e x i s t e n c e of an L i - m e t r i c on Dj ( i n d u c e d by the L x - n o r m on the l i n e a r space spanned by D^) so that the l i t t l e o term i n (1.11) is w e l l - d e f i n e d . Machina ensures this by r e s t r i c t i n g d i s t r i b u t i o n s under cons iderat ion to those with supports in some compact i n t e r v a l , say [0,M] . We denote by D ^ Q J^-J the set of d i s t r i b u t i o n s so r e s t r i c t e d . The rest of this essay w i l l develop some r i s k aversion impl icat ions of mainly l i n e a r Gateaux u t i l i t y and weighted u t i l i t y . Given the wide- spread acceptance of expected u t i l i t y in finance and economics, we w i l l present the resu l t s of expected u t i l i t y as a benchmark for comparisons. - 18 - Before we formally introduce the representat ions of various p r e f e r - ence theor ies , a few c l a r i f i c a t i o n s on some assumptions i m p l i c i t l y made and terms and notations not formally defined w i l l avoid confus ion. F i r s t of a l l , whi le Frechet d i f f e r e n t i a b l e u t i l i t y is defined on D|-Q , expec- ted u t i l i t y and weighted u t i l i t y in p a r t i c u l a r and l i n e a r Gateaux u t i l i t y i n general can be extended to non-numerical outcome space. Since th is essay takes a finance perspect ive and focuses on monetary outcomes, we w i l l only present these u t i l i t y theories with numerical outcomes. In other words, the von Neumann-Morgenstern u t i l i t y function of an expected u t i l i t y dec i s ion maker and both the value and the weight functions of a weighted u t i l i t y dec i s ion maker are assumed to be J •*• R mappings, where J i s a subset of R. We w i l l denote by 6 x the step d i s t r i b u t i o n funct ion with a l l mass centered at point x. In other words, 6 stands for the l o t - r x tery of ge t t ing x for sure. Sometimes we need to specify the random v a r i - able of a d i s t r i b u t i o n . In such cases, F ~ is used to denote the d i s t r i b u - ' x t i o n of random v a r i a b l e x. S e c o n d l y , we use "V1 to denote a p r e f e r e n c e o r d e r i n g on Dj (or Drr> M l ' depending on the c ircumstances) . For F , G in D , "F > G" means F i s weakly preferred to G. When F > G and G > F , we say F and G are i n d i f - f e r e n t , denoted by "F ~ G". I f F > G and not F ~ G, we say F is s t r i c t l y preferred to G, denoted by "F >- G". T h i r d l y we use the fol lowing labe l s to shorten our statements: EU expected u t i l i t y WU weighted u t i l i t y LIU l i n e a r i m p l i c i t u t i l i t y - 19 - FDU Frechet d i f f e r e n t i a b l e u t i l i t y LGU l i n e a r Gateaux u t i l i t y At times, we w i l l re fer to a dec i s ion maker by his preference-represent ing funct ions . For example, we might re f er to an expected u t i l i t y dec i s ion maker with von Neumann-Morgenstern u t i l i t y funct ion u as 'EU dec i s ion maker u ' , a weighted u t i l i t y dec i s ion maker with value funct ion v and weight funct ion w as 'WU dec i s ion maker (v ,w) ' , an FDU dec i s ion maker with l o c a l u t i l i t y functions u(x;F) as 'FDU dec i s ion maker u ( x ; F ) ' and an LGU d e c i s i o n maker whose preference funct iona l V s a t i s f i e s condi t ion (1.2) as 'LGU dec i s ion maker C ( x ; F ) ' . Or , we may i d e n t i f y a dec i s ion maker by his preference f u n c t i o n a l , i . e . EU for expected u t i l i t y , WU for weighted u t i l i t y , and V for FDU and LGU. F i n a l l y , the terms 'decreas ing ' , ' i n c r e a s i n g ' , 'concave' , 'convex 1 , e t c . , are used in the weak sense. When the s t r i c t sense a p p l i e s , i t w i l l be obvious by context or so i n d i c a t e d . 1.3 Representation Hypothesis EU1: There ex is t s a continuous, increas ing funct ion u: J -> R such that , for any F , G £ D j , EU(F) > EU(G) <=> F > G where EU(F) = /u(x)dF(x) . (1.14) It is known that the u t i l i t y funct ion u in Hypothesis EU1 exis ts i f and only i f the preference ordering s a t i s f i e s the fol lowing axioms: - 20 - Axiom 1 (Completeness): For any F , G e D T , e i ther F > G or G > F . j ~ ~ Axiom 2 ( T r a n s i t i v i t y ) : For any F , G, H e D T , i f F > G and G > H, then F > j ~ ~ ~ H. Axiom 3 ( S o l v a b i l i t y ) : For any F , G, H e D j , i f F > G > H, then there ex i s t s a 6 e (0,1) such that 8F+(1-8)H ~ G. Axiom 4 ( M o n o t o n i c i t y ) : For any F , G e D J } i f F > G and 1 > 8 > y > 0, then BF+(1-8)G > yF+( l -Y)G. Axiom 5 ( S u b s t i t u t i o n ) : For any F , G, H e D , and p e [0 ,1 ] , i f F ~ G, J then pF+(l-p)H ~ pG+(l-p)H. Moreover, any u and u* s a t i s f y i n g the r e l a t i o n s h i p u* - a + bu, b > 0 (1.15) are equivalent representat ions for a preference o r d e r i n g . The f i r s t four axioms of expected u t i l i t y are innocuous and norma- t i v e l y appeal ing . Axiom 4 i s sometimes c a l l e d 'mixture-monotonic i ty' (Chew, 1983; 1984; F i shburn , 1983), and i s equivalent to a property c a l l e d 'betweenness' (Chew, 1983). D e f i n i t i o n 1 .1 : A p r e f e r e n c e o r d e r i n g > on Dj i s s a i d to d i s p l a y the betweenness property i f , for any F , G e D T s a t i s f y i n g F > G and F a = (1- a)F+aG with a e (0 ,1 ) , i t i s always true that F > F a > G. In words, betweenness means that any p r o b a b i l i t y mixture of two l o t - t e r i e s must be intermediate i n preference between them. I f F ~ G, then F ~ F a ~ G. Since F a = (l-a)F+aG for any a e (0,1) l i e s on the l i n e segment connecting F and G, the betweenness property impl ies that the agent's i n - d i f f erence curves i n any simplex of 3-outcome l o t t e r i e s must be s t ra ight l i n e s . - 21 - The s u b s t i t u t i o n axiom has been c o n t r o v e r s i a l and is the primary cause for the preference funct ional EU to be l i n e a r in d i s t r i b u t i o n . It impl ies that the agent's ind i f f erence curves in the above-mentioned sim- plex must furthermore p a r a l l e l each other. Many attempts to general ize expected u t i l i t y have aimed at re lax ing th i s axiom. We mentioned e a r l i e r that Machina assumes Frechet d i f f e r e n t i a b i l i t y to do away with the strong s u b s t i t u t i o n axiom. How weighted u t i l i t y theory does i t w i l l be e labora- ted s h o r t l y . The funct ion u in (1.14) is usua l ly re ferred to as the (von Neumann- Morgenstern) u t i l i t y funct ion . Re la t ionsh ip (1.15) is the a f f ine t rans - formation that defines the uniqueness c lass of u . Weighted u t i l i t y theory is an axiomatic genera l i za t ion of expected u t i l i t y advanced by Chew and MacCrimmon (1979a; 1979b), Chew (1980; 1981; 1982; 1983), Fishburn (1983) and Nakamura (1984). L ike expected u t i l i t y , and d i s t i n c t from other a l t e r n a t i v e approaches mentioned in Sect ion 0, i t i s derived from a set of assumptions about the underlying preferences . It re ta ins the completeness, t r a n s i t i v i t y , s o l v a b i l i t y and monotonicity axioms of expected u t i l i t y theory, but weakens i t s (strong) s u b s t i t u t i o n axiom v i a the fo l lowing: Axiom 5 ' (Weak S u b s t i t u t i o n ) : For any F , G e Dj such that F ~ G and any 8 e (0 ,1 ) , there ex is t s aye (0,1) such that , for every H e D j S BF+(1-8)H ~ Y G + ( I - Y ) H . Axiom 5' d i f f e r s from Axiom 5 in al lowing y t B. y and 8 however must s t i l l s a t i s f y a r e l a t i o n s h i p c a l l e d ' r a t i o cons i s t ency 1 : D e f i n i t i o n 1.2: For any F , G, H E D j ( and 8 ,̂ fL,, Y]_> Y2 e (0,1) such that F ~ G and B.F+(1-B.)H ~ Y-G+(1"Y-)H for i = 1, 2, i f 1 1 1 1 - 22 - y/U-y) Y 9 / ( l - Y 9 ) " (1.16) Pj/d-pj) P 2 / ( I - P 2 ) ' then, we say the preferences exhibit the r a t i o consistency property. A proof of the following lemma appears i n Chew (1983, Lemma 2). Lemma 1.1; Axioms 1, 2, 4 and 5' imply r a t i o consistency. In a simplex of l o t t e r i e s involving three outcomes x, x, x e J, with x < x < x as i l l u s t r a t e d i n Figure 1.1, suppose P is the p r o b a b i l i t y mix- ture of 6 and 6- such that 6 ~ P . Betweenness and r a t i o consistency to- x x x J gether implies that the indifference curves must be straight lines which 'spoke out' from a point, say A, on the l i n e connecting 6 x and P. (A must be to the right of 6̂ or to the l e f t of P, i . e . outside of the simplex, or t r a n s i t i v i t y w i l l be v i o l a t e d . ) Hypothesis WU1: There exist a s t r i c t l y increasing function v: J ->• R and a s t r i c t l y p o s i t i v e function w: J •> R such that, for any F, G e Dj, WU(F) > WU(G) <=> F K , where WU(F) • MfXMffiF$X) . (1.8) Jw(x;dF(x; The v in (1.8) i s referred to as the value function and w the weight function. Suppose another pair of value and weight functions (v*,w*) also represent the same preference ordering. Then v, v*, w and w* must s a t i s f y the following uniqueness-class transformation r e l a t i o n s h i p s : v * - f£f (1.17) w* = (sv+t)w (1.18) where q, t, r and s are constants s a t i s f y i n g qt > rs and sv+t > 0. 23 - Figure 1.1: Indifference curves in a simplex of lotteries involving three outcomes x < x < x - 24 - Note from (1.8) that WU(F) is not l i n e a r in F and can be rewri t ten as: WU(F) = jv (x )dF W (x ) , (1.19) where F w ( x ) = — . (1.20) / *w(t)dF(t) J — CO J + ° ° w ( t ) d F ( t ) C l e a r l y , when w is constant, WU w i l l reduce to EU since F W (x) = F ( x ) . Because Machina's FDU analys is is not an axiomatic approach, there is not a representat ion theorem. For ease of comparison, we restate the approach Machina proposed as fo l lows: Hypothesis FDU1: There ex is t s a Fre*chet d i f f e r e n t i a b l e preference func- t i o n a l V: D J Q J^-J -> R such that V(G) - V(F) = /u(x;F)d[G-F] + oIlG-Fll. (1.21) Machina c a l l e d the Frechet d e r i v a t i v e u ( x ; F ) : [0,M] •*• R a l o c a l u t i l i t y funct ion at d i s t r i b u t i o n F . To obtain tes table i m p l i c a t i o n s , Machina further assumed the fo l low- ing s p e c i f i c form: V(F) = /R( t )dF( t ) ±j [ / s ( t )dF( t ) ] 2 (1.22) with l o c a l u t i l i t y funct ion u(x;F) = R(x) ± S(x) Js ( t )dF( t ) . (1.23) This 'quadrat ic in p r o b a b i l i t y 1 funct iona l is known to be incompatible with the betweenness property . In a simplex of 3-outcome l o t t e r i e s , th is means that the agent's ind i f f erence curves are i n general not s t ra ight l i n e s . - 25 - Note that the outcome space in Hypothesis FDU1 is D J Q ^ while that i n both Hypotheses EU1 and WU1 is D^. When r e s t r i c t e d to ^ , both EU and WU are also Frechet d i f f e r e n t i a b l e . Furthermore, the funct iona l EU in Hypothesis EU1 has a constant (with respect to d i s t r i b u t i o n s ) Fre*chet d e r i v a t i v e u which does not depend on d i s t r i b u t i o n F so that the term oilG— FII in expression (1.21) vanishes . The requirement of a compact outcome space means that Machina's approach might not be extendable to l o t t e r i e s with non-compact supports. Since LGU contains FDU as a spec ia l case and can allow unbounded outcome space, 4 we s h a l l adopt LGU as the most general preference funct iona l to be discussed in th is essay. Later i n Section 6, we w i l l need to impose more s tructure on the l i n e a r Gateaux d e r i v a t i v e C of an LGU funct ional in order for Machina's r e su l t s to hold under LGU. Hypothesis LGU1: There ex i s t s a preference funct iona l V: Dj •*• R such that V(F t t ) = / C ( x ; F a ) d [ G ( x ) - F ( x ) ] , (1.2) a <x v where F " = ( l -a)F+aG and C( •; •): J * Bj ->• R. We w i l l c a l l C(x;F) the l o t t e r y s p e c i f i c ( w . r . t . F) u t i l i t y funct ion (LOSUF) of V . It should be pointed out that both FDU and LGU are so general that one might f ind them lack ing in s t r u c t u r a l c o n s t r a i n t s . For instance , i t i s not known what transformation defines the uniqueness class for FDU or LGU type preferences . Once we introduce c e r t a i n t y equivalent r i s k avers ion , the support J is required to be bounded from below. See Section 2. - 26 - We next examine the condit ions needed for each preference funct ional to d i sp lay the normatively appealing property c a l l e d ' s tochas t i c dominance'. 1.4 Stochastic Dominance It is genera l ly agreed that any preference ordering should be consistent with s tochast ic dominance defined below: D e f i n i t i o n 1 .3: For F , G E D J ( F is said to s t o c h a s t i c a l l y dominate G in the f i r s t degree, denoted by F > l G, i f F(x) < G(x) for a l l x e J . I f moreover F(x) < G(x) for some x e J , then F is said to s t r i c t l y stochas- t i c a l l y dominate G in the f i r s t degree, denoted by F > 1 G. G r a p h i c a l l y , s tochast ic dominance in the f i r s t degree means that F and G do not cross and F always l i e s below ( i . e . to the r ight of) G. D e f i n i t i o n 1.4: A p r e f e r e n c e o r d e r i n g > i s s a i d to be consistent with s tochast ic dominance (SD) i f F > G whenever F > i G. In other words, i f F always de l i ver s a better outcome with a higher p r o b a b i l i t y than G, then F ought to be preferred to G. It is easy to check the fo l lowing: Lemma 1.2: I f F >i G and F " = ( l -a)F+aG, then F > i F™ >L F " ' >l G for any a, a' e (0,1) such that a < a1 . Therefore , regardless of the underlying preference theory, Axiom 4 (Monotonicity) implies that V(F t t ) must decrease in a i f V is to be cons i s - tent with SD. Theorem U2 (SD) : For F , G £ D J ( F a = (l-a)F+aG where a e (0 ,1 ) , and any preference funct iona l V: Dj •* R s a t i s f y i n g Assumption 1.1, - 27 - G implies V(F) > V(G) i f and only i f F > i G implies ^ - V C F " ) < 0. d a When V is an EU f u n c t i o n a l , d_ d a EU (F a ) - lim i {EU(F a + 9) - EU(F a) } e - o = lim 4 {/u(x)d[(l-a-9)F+( a+0)G] - Ju(x)d[(1-a)F+aG] } e-K) = lim i {/u(x)dFa+ 6/u(x)d[G-F] - Ju(x)dF a} 6-K) = /u(x)d[G-F] = -Ju'(x)[G(x)-F(x)]dx = Ju ,(x)[F(x)-G(x)]dx. Since F(x)-G(x) < 0 by the d e f i n i t i o n of > i and u'(x) > 0 by Hypothesis EU1, i t i s always true that EU is consistent with SD. Theorem EU2 (SD): Suppose u: J -*• R i s continuous and i n c r e a s i n g . Then, for any F , G £ D j , F > i G implies EU(F) > EU(G) . "u' > 0" is commonly re ferred to as the necessary and s u f f i c i e n t cond i t ion for EU to be consistent with SD. Since under EU, u is by cons truc t ion an increas ing funct ion , Theroem EU2 stresses that the preferences of an EU dec i s ion maker are consistent with SD. When V is a WU f u n c t i o n a l , r e c a l l that the Gateaux d i f f e r e n t i a l of WU(F a) i s dWU(F a) = JC (x ;F a )d [G(x) -F(x ) ] (1.9) d a = - / [ G ( x ) - F ( x ) ] d C ( x ; F a ) = -/C ,(x ; F a ) [ G(x ) - F(x)]dx. (by in t egra t ion by parts) (1.24) - 28 - I f F > i G, then G(x)-F(x) > 0 for a l l x e J by d e f i n i t i o n . The nece- s s a r y and s u f f i c i e n t c o n d i t i o n for : < 0 is therefore C ' (x:F) > 0 J da for a l l x e J at a l l F e D j . Theorem WU2 (SD): Suppose w, v are continuous and bounded; v i s s t r i c t l y increas ing and w is s t r i c t l y p o s i t i v e . Then, for any F , G £ D j , F > l G implies WU(F) > WU(G) i f and only i f C(x;F) = w(x)[v(x)-WU(F)]//wdF (1.25) i s an increas ing function of x for a l l F e Djj or , e q u i v a l e n t l y , <t>(x,s) = w(x)[v(x)-v(s) ] (1.26) is an increas ing funct ion of x for a l l s £ J . Confirming Theorem U2, the condi t ion for SD under WU is that the Gateaux d i f f e r e n t i a l of WU(F) be decreasing in the d i r e c t i o n G - F . R e c a l l that , under EU where u'(x) > 0 guarantees i t s consistency with SD, u(x) also has the funct iona l a n a l y t i c a l i n t e r p r e t a t i o n of a Gateaux d e r i v a t i v e , but does not depend on the d i s t r i b u t i o n F because the EU representat ion EU(F) = Ju(x)dF(x) i s l i n e a r in d i s t r i b u t i o n . This observation led Chew and MacCrimmon to name C(x;F) a ' l o t t e r y s p e c i f i c u t i l i t y funct ion (LOSUF)' . We also apply the use of th is term to the Gateaux d e r i v a t i v e of an LGU f u n c t i o n a l . As such, Machina's ' l o c a l u t i l i t y funct ion' i s also a LOSUF. We avoid using the term ' l o c a l ' here since i t s meaning is d i f f e r e n t from the ' l o c a l ' in ' l o c a l r i s k a v e r s i o n ' , to be discussed in the next s ec t ion . In Theorem WU2, the condi t ion for SD is given in terms of both C ( x ; F ) : J x Dj -*• R and <t>(x,s): Jz + R. Since /wdF is constant given F , condit ions on C(x;F) and ())(x,s) are equivalent . As <|)(x,s) i s d i s t r i b u - t i o n - f r e e , i t may at times of fer more i n t u i t i v e i n t e r p r e t a t i o n s . The condi t ion for SD i n terms of <|>(x,s) can be wr i t ten as: ^ ( x . s ) = w' (x) [v(x) -v(s ) ] + w(x)v'(x) > 0, (1.27) which in turn can be rewri t ten as: w'(x) > v ' (x ) , < / . o o N / s , 7—T 7—v f o r a 1 1 s * x. (1.28) w(x; < v ( x ; - v ( s ; > When [ lnw(x)]' e x i s t s , (1.28) is equivalent to: [ lnw(x)] ' I - / s ( x j v for a l l s $ x. (1.29) < v (x ; -v ( s ) > Let v = max | v (x ) } and v = min (v(x) }. Condit ion (1.29) can then be rewri t ten as ^ r - = t lnw(x)] ' < V ' ( x ) i f w'(x) > 0, (1.30) v -v (x ; = [ lnw(x)]' > - y ' ( , x ) i f w'(x) < 0. (1.31) w(x) v (x ) -v In other words, SD requires the rate of change of lnw(x) be bounded from above and from below by the RHS of (1.30) and (1 .31) , r e s p e c t i v e l y — i . e . , when w(x) is i n c r e a s i n g , lnw(x) cannot increase too fas t ; when w(x) is decreasing, lnw(x) cannot decrease too r a p i d l y e i t h e r . In the above, we considered mul t ip l e d i s t r i b u t i o n s . The SD condi t ion is to guarantee that F w i l l be preferred to G as long as F > l G. If we are concerned with only one p a r t i c u l a r d i s t r i b u t i o n , what is the meaning of C being increas ing? Note from expression (1.25) that C(x;F) has the i n t e r p r e t a t i o n of a "weighted u t i l i t y - d e v i a t i o n from WU(F)" with w(x)//wdF being the weight. - 30 - It i s therefore not s u r p r i s i n g that /C(x;F)dF(x) = 0. The d e r i v a t i v e r , , . _ w'(x)[v(x)-WU(F)]+w(x)v'(x) C (x ,F) = 7 w o f I [ v ( x ) - W U ( F ) ] + 7 w l F - V ' ( x ) ( 1 - 3 2 ) i s accordingly a "marginal weighted u t i l i t y - d e v i a t i o n from WU(F)" — the increase in C(x;F) caused by an i n f i n i t e s i m a l increase of x, and is the combined effect of two forces represented by the two terms in (1 .32) . G i v e n a d i s t r i b u t i o n F , suppose x increases margina l ly to x + . WU(F) and /wdF are constant. w(x) and v(x) may be viewed as unchanged. v ' (x ) is s t r i c t l y p o s i t i v e . w'(x) may be p o s i t i v e or negative depending on whether (v,w) is o p t i m i s t i c or pess imis t i c at x. According to (1 .32) , when x i n - creases , i t causes two ef fects on £ . F i r s t , i t changes the weight w(x). Second, i t changes the contingent 'sure u t i l i t y ' v ( x ) . In (1 .32) , the f i r s t term gives the change of C r e s u l t i n g from the change in weight, holding the contingent u t i l i t y - d e v i a t i o n at i t s i n i t i a l l e v e l . The second term gives the p o s i t i v e ef fect on C caused by the increase in the c o n t i n - gent u t i l i t y , assuming i t s weight has not changed. There are four p o s s i - ble cases as l i s t e d below: case v(x)-WU(F) ^ g l ^ g l [ v ( , ) - w P ( F ) ] ^ v ' ( x ) C' (x ;F) (1) + + + + + (2) + - - + ? (3) - + - + ? (4) - - + + + - 31 - Obviously , cases (1) and (4) pose no ambiguity. To also have C ' (x;F) > 0 in both case (2) and case (3) requires that the agent be not exces- s i v e l y pess imis t i c when x is better than his c e r t a i n t y equivalent of the d i s t r i b u t i o n , and not ' o v e r l y - o p t i m i s t i c ' e i ther when x is below the c e r - t a i n t y equiva lent . We w i l l repeatedly see l a t e r that , as far as preference propert ies are concerned, C(x;F) is the WU equivalent of the von Neumann-Morgenstern u t i l i t y function u(x) . I f th is i s to make sense, u(x) must also be able to give a u t i l i t y - d e v i a t i o n i n t e r p r e t a t i o n . This i s obviously true in l i g h t of the af f ine EU uniqueness transformation although i t is r a r e l y so in terpreted i n the l i t e r a t u r e . For LGU, s i n c e the Gateaux d e r i v a t i v e of V ( F a ) at F is C (x ;F) , the fol lowing is true: Theorem LGU2 (SD): Let V: Dj •*• R be a l i n e a r Gateaux preference funct ional with LOSUF C: R * Dj •*• R. Then, F G implies V(F) > V(G) i f and only i f C(x;F) is increas ing i n x for a l l F e D j . Given the p l a u s i b i l i t y of SD, we w i l l consider only the functions that s a t i s f y the required cond i t ions . - 32 - 2 INDIVIDUAL RISK AVERSION In f i n a n c i a l economics, we often assume that dec i s ion makers are r i s k averse. The notion of r i s k aversion can however be defined d i f f e r e n t l y based on d i f f e r e n t concepts. For example, i f an agent's c e r t a i n t y equiva- lent for any l o t t e r y is always less than the expected value of that l o t t e - r y , we may say that he is r i s k averse in the c e r t a i n t y equivalent sense. A l t e r n a t i v e l y , i f the insurance premium an agent is w i l l i n g to pay to trade away an a r b i t r a r y r i s k is always greater than the mean of that r i s k , then we may say that he is r i s k averse i n the sense of insurance premium. We can even define r i s k aversion in terms of an agent's asset demand or his subject ive value of informat ion , e t c . In another d i r e c t i o n , i t is often of interes t to d i s t i n g u i s h agents' r i s k at t i tudes ' i n the smal l ' and ' i n the l a r g e ' . The oberservat ion that people do hold insurance p o l i c i e s and l o t t e r y t i cke t s s imultaneously c l e a r l y suggests that people have d i f f e r e n t a t t i tudes towards r i s k s of d i f f e r e n t ' s i z e s ' . Given a wealth p o s i t i o n , an agent's r i s k aversion to - wards i n f i n i t e s i m a l r i s k s is convent ional ly termed l o c a l r i s k avers ion . In contras t , his r i s k aversion towards r i sks in general is c a l l e d g lobal r i s k avers ion . Under expected u t i l i t y , however, these d i f f erent r i s k aversion no- t ions are a l l equivalent to the concavity of the u t i l i t y funct ion . Since some of these r i s k aversion notions are not equivalent under WU and LGU, i t is necessary that we make d i s t i n c t i o n s between them. We w i l l begin - 33 - with l o c a l r i s k avers ion . 2.1 Loca l Risk Avers ion In the l i t e r a t u r e of r i s k avers ion , l o c a l r i s k aversion refers to r i s k a v e r s i o n towards small r i s k s . Suppose e i s an a r b i t r a r y i n f i n i t e s i - mal , a c t u a r i a l l y f a i r r i s k and the dec i s ion maker's wealth l e v e l is x. I f the dec i s ion maker always prefers h is status quo to taking r i s k e ( i . e . 6^ >̂ F x + ^ ) > we would l i k e to say that h i s p r e f e r e n c e s d i sp lay l o c a l r i s k avers ion at x. While the r i s k being considered here has to be i n f i n i t e s i - m a l , i t need not be a c t u a r i a l l y f a i r . Suppose E( e) * 0 and the agent can pay a premium to insure against th i s r i s k . As long as the premium is g r e a t e r than E ( e ) , i t seems reasonable to say that th is agent is averse towards the small r i s k e. To define l o c a l r i s k aversion formal ly , we f i r s t define the term ' insurance premium'. D e f i n i t i o n 2.1: I f a d e c i s i o n maker i s i n d i f f e r e n t between F , ~ and ; x+e 6 _ / ~ \ , then % i s c a l l e d h i s (uncondit ional) insurance premium for x + E ( e ; - i i c r i s k e at x. We also define ' c o n d i t i o n a l insurance premium' which w i l l be needed for any n o n l i n e a r - i n - d i s t r i b u t i o n preference theories such as WU and LGU. D e f i n i t i o n 2.2: I f a d e c i s i o n maker is i n d i f f e r e n t between p F x + ^ + ( l - p ) H and p 6 + ( l - p ) H , then % is ca l l ed his c o n d i t i o n a l insurance pre - v x + E ( e ) - n K ' - mium for r i s k e at x c o n d i t i o n a l on p and H. F i r s t , note that the r i s k e i n De f in i t i ons 2.1 and 2.2 is an a r b i t r a - ry r i s k which need not be a c t u a r i a l l y f a i r or i n f i n i t e s i m a l . Second, the - 34 - x i n D e f i n i t i o n s 2.1 and 2.2 can be in terpreted as the dec i s ion maker's sure weal th p o s i t i o n p r i o r to t a k i n g the r i s k e. In g e n e r a l , u w i l l depend on x, E , and the i n d i v i d u a l ' s a t t i tudes towards r i s k . D e f i n i t i o n 2 . 3 : I f a d e c i s i o n maker's insurance premium for any r i s k e, i t ( x , e ) , i s p o s i t i v e , then h i s preference is sa id to d i sp lay (uncondi- t i o n a l ) insurance premium r i s k avers ion (IPRA) at x. D e f i n i t i o n 2.4: If a dec i s ion maker's c o n d i t i o n a l insurance premium for any r i s k e, n(x , £ j p , H ) , i s p o s i t i v e for any p £ (0,1] and H £ Dj i t i s c o n d i t i o n a l upon, then his preference i s sa id to d i sp lay c o n d i t i o n a l insurance premium r i s k avers ion (CIPRA) at x. L o c a l r i s k avers ion i s a s p e c i a l case of uncondi t iona l IPRA because i t r e s t r i c t s the r i s k s under cons iderat ion to the i n f i n i t e s i m a l ones. D e f i n i t i o n 2.5: If a dec i s ion maker's insurance premium for any i n f i n i t e - s i m a l r i s k e, 7 t ( x , £ ) , i s p o s i t i v e , then his preference i s sa id to d i s - play l o c a l r i s k avers ion (LRA) at x. I f his preference d i sp lays LRA at a l l x, we say that i t d i sp lays pointwise LRA (PLRA). C l e a r l y , IPRA implies LRA. We could conceivably define a term c a l l e d ' c o n d i t i o n a l PLRA' . This however w i l l not be considered i n t h i s essay. We s a i d p r e v i o u s l y that n depends on x, £ and the agent's r i s k a t t i - t u d e s . Assume that the v a r i a n c e of £ i s a1. How can we express n i n terms of E, X and the a g e n t ' s u t i l i t y funct ion? The now famous Arrow- Prat t index provides a convenient way. D e f i n i t i o n 2.6: A funct ion r : J -> R i s an Arrow-Pratt index of a p r e f e r - ence ordering i f the (uncondi t ional ) insurance premium T t ( x , £ ) for an i n - f i n i t e s i m a l r i s k £ with variance az •*• 0 can be wri t ten as TI (X , £ ) = ^ r ( x + E ( £ ) ) + o(az). (2.1) Given D e f i n i t i o n 2 .5 , C o r o l l a r y 2.1 below i s obvious. C o r o l l a r y 2 .1: A dec i s ion maker with Arrow-Pratt index r(x) i s LRA at x i f and only i f r (x) > 0. He i s PLRA i f and only i f r (x) > 0 at a l l x e J . The fo l lowing theorems are wel l known: Theorem EU3 (Arrow-Pratt Index): The Arrow-Pratt index of an EU dec i s i on maker with a continuous, s t r i c t l y i n c r e a s i n g , t w i c e - d i f f e r e n t i a b l e von Neumann-Morgenstern u t i l i t y funct ion u(x) i s given by r (x ) = - • (2.2) u'(x) v 7 Theorem EU4 (PLRA): The preference of an EU dec i s ion maker u w i l l d i sp lay PLRA i f and only i f u i s concave. For a WU dec i s i on maker (v,w), l e t it be his insurance premium for an i n f i n i t e s i m a l , a c t u a r i a l l y f a i r r i s k e with small variance a2. It can be shown that , at wealth p o s i t i o n x, / ~ \ ° z r v"(x) , 2w'(x), , „ •>* , „ „ . H - u(x,e) - - - j . [ _ r y - + _ ^ i . ] + 0 < ^ ) . (2.3) Theorem WU3 (Arrow-Pratt Index): The Arrow-Pratt index of a WU dec i s ion maker with proper ly s tructured value funct ion v and weight funct ion w i s given by r(x) = - [ l £ E > + 2 w ; ( x ) ] = - ^4 - ̂ L . (2.4) L v ' (x ) w(x) J v ' (x ) w(x) v ' I t i s worth noting that , l i k e i t s expected u t i l i t y counterpart , the WU Arrow-Pratt index r(x) i n (2.4) i s invar iant under the uniqueness c lass transformations (1.17) and (1 .18) . Express ion (2.4) suggests that a WU dec i s ion maker's avers ion toward small r i s k s can be seen as coming from two sources represented by two a d d i t i v e terms. The f i r s t term - v " / v ' can be c a l l e d the 'value-based r i s k avers ion index' which measures r i s k aversion a t t r i b u t a b l e to the value - 36 - funct ion v . The second term -2w'/w can be in terpreted as the 'percept ion- based r i s k aversion index' or simply the 'optimism (pessimism) index' that r e f l e c t s c e r t a i n q u a l i t i e s of the dec i s ion maker's percept ion about the prospects i n question (Weber, 1982). To d i sp lay PLRA, the sum of these two components must be p o s i t i v e at a l l x. The concavity of v alone i s ne i ther necessary nor s u f f i c i e n t for PLRA. When the weight funct ion i s constant , EU r e s u l t s , and r(x) reduces to the t r a d i t i o n a l Arrow-Pratt index. By C o r o l l a r y 2 .1 , the preference of a WU dec i s ion maker (v,w) w i l l d i s p l a y LRA at x i f and only i f r (x) given by (2.4) i s p o s i t i v e . Given that r ( x ) = - [ 1 ^ 0 + 2 w J W j ( 2 . 4 ) Lv (x) w(x) ' = - { ln [v ' (x )w 2 (x ) ] }', (2.5) the fo l lowing i s obvious: Theorem WU4 (PLRA): The preference of a WU dec i s ion maker (v,w) w i l l d i s - play PLRA i f and only i f ln [v ' (x )w 2 (x ) ] i s decreasing i n x. Under EU, the von Neumann-Morgenstern u t i l i t y funct ion can be reco- vered from the Arrow-Pratt index r(x) uniquely (up to an a f f ine t r a n s f o r - mation) v i a the fo l lowing: u(x) = / exp[ - / r (x )dx ]dx . (2.6) Therefore , two EU maximizers who share the same Arrow-Pratt index must have the same u t i l i t y f u n c t i o n . From (2.5) above, i t i s c l ear that , under WU, what we can recapture from the Arrow-Pratt index i s v ' ( x ) w 2 ( x ) . Therefore , i t i s poss ib le that two d i s t i n c t pa irs of value and weight functions share the same Arrow- Prat t index and exh ib i t i d e n t i c a l l o c a l r i s k p r o p e n s i t i e s . - 37 - The condit ions i n Theorems EU4 and WU4 for PLRA are both necessary and s u f f i c i e n t . I t w i l l be i n t e r e s t i n g . to know what more s p e c i f i c condi - t ions are s u f f i c i e n t for WU to d i sp lay PLRA. C o r o l l a r y 2.2 below i d e n t i - f i e s two such c o n d i t i o n s . C o r o l l a r y 2.2: The preference of a WU dec i s ion maker (v,w) w i l l d i sp lay PLRA i f condi t ion ( i ) or ( i i ) below holds: ( i ) w i s constant and v i s concave; ( i i ) v i s l i n e a r and w i s decreas ing . Proof: Omitted. In case ( i ) where w i s constant, WU reduces to EU. Consequently, v being concave i s necessary and s u f f i c i e n t for PLRA by Theorem EU4. Under cond i t ion ( i i ) where v i s l i n e a r , the f i r s t term of r(x) i n (2.4) vanishes and a decreasing w w i l l r e s u l t i n a p o s i t i v e Arrow-Pratt index. To character ize LRA for an LGU dec i s ion maker V , we must f i r s t derive h i s A r r o w - P r a t t i n d e x . Suppose E i s an i n f i n i t e s i m a l , a c t u a r i a l l y f a i r r i s k with small variance o 2 . Then, by the d e f i n i t i o n of insurance premium, V < W = V< FX+E>- Let F a = ( l - a ) F ~ + a6 . We have X+E X-1X 0 = V( 6 J - V(F ~) = t da x - i r X + E J0 da = / J { / C ( s ; F a ) d [ 6 x _ i i - F x + ~ ] } d a (from (1.2)) = / { / J c C s j F V a J d ! C ^ - F x + ~ ] = / C ( s ; F a ' ) d [ 6 x _ 7 t - F x + ~ ] for some a' £ (0 ,1 ) . Hence, - 38 - C ( x - i i ; F a ' ) = /C (x+s;F a ' )dF~ . ^ OC' ') Noting that e i s a small r i s k and that F •> § x as o z + 0, we can take the T a y l o r ' s expansion for both sides as fo l lows: C ( x - n ; F a ' ) - C(x-7t;6 ) = C(x; 6 ) - it C (x; 6 ) + 0 ( i t 2 ) ; and /C (x+s ;F a ' )dF~ - /C(x+s;6 )dF- = J [ C ( x ; 6 x ) + s C ' ( x ; 6 x ) + 1^ C " ( x ; 6 x ) + o ( s 2 ) ] d F ; = C ( x ; 6 x ) + ̂ C " ( x ; 6 x ) + o ( a 2 ) . Therefore , CT2 C " ( x ; 6 x ) a2 it = it(x, e ) = _ [ - . 5 ] + o ( a 2 ) = r ( x ) + o ( a 2 ) , (2.7) > x where C * ( x ; 6 x ) r ( x ) = " C ' ( x ; 6 x ) * ( 2 , 8 ) When E(e) * 0, expression (2.7) becomes „ v C M (xt-E(e);6 ~ ) 7t = Tt(x, e) = ^ [- * + E { e ) ] 4- o ( o 2 ) . (2.9) C ' ( x + E ( e ) ; 6 x + E ( ~ ) ) Hence, Theorem LGU3 (Arrow-Pratt Index): The Arrow-Pratt index of an LGU dec i s ion maker V with continuous, s t r i c t l y i n c r e a s i n g , t w i c e - d i f f e r e n t i a b l e LOSUF C(x;F) i s given by expression (2.8) above. Theor em LGU4 (PLRA): The preference of an LGU dec i s i on maker' V with LOSUF C(x;F) w i l l d i sp lay PLRA i f C(x;F) i s concave i n x for a l l F . Note that , the condit ions of u being concave i n Theorem EU4 and l n [ v ' w 2 ] being decreasing i n Theorem WU4 are both necessary and s u f f i c i e n t - 39 - while the concavity of C(x;F) i n Theorem LGU4 i s only s u f f i c i e n t for PLRA. 2.2 G l o b a l Risk Avers ion In contrast to l o c a l r i s k avers ion , g loba l r i s k avers ion (GRA) refers to r i s k aversion i n the l a r g e . I f two d i s t r i b u t i o n s F and G share the same mean and G has a higher v a r i a b i l i t y , then we would expect a r i s k averse agent to prefer F to G. Global r i s k avers ion i s such a concept. Given the prevalence of various forms of l o t t e r i e s throughout the world, i t i s perhaps u n r e a l i s t i c to require a l l agents not to have p r e f e r - ence for any a c t u a r i a l l y u n f a i r r i s k s — an i m p l i c a t i o n of g loba l r i s k avers ion . I t i s , however, des i rab le for a u t i l i t y theory, be i t l i n e a r or nonl inear i n d i s t r i b u t i o n , to be able to d i sp lay some form of g loba l r i s k aversion when a s p e c i f i c a p p l i c a t i o n context c a l l s for i t . Af ter Rothschi ld and S t i g l i t z (1970), g lobal r i s k avers ion in finance i s better known v i a 'mean preserving s p r e a d 1 . Instead of jumping into the d e f i n i t i o n of mean preserving spread, we s tar t with a less general , but s impler and more i n t u i t i v e concept which we c a l l 'simple mean preserving spread ' . D e f i n i t i o n 2.7: For F if G, G i s sa id to s i n g l e - c r o s s F at x* from the l e f t i f G(x) -F(x) > 0 for a l l x < x* (2.10) and G(x)-F(x) < 0 for a l l x > x*. (2.11) When there i s no ambiguity about the d i r e c t i o n , we say that G and F possess the s ing le cross ing property . - 40 - D e f i n i t i o n 2.8: G i s a simple mean preserving spread (simple mps) of F i f (a) G single-crosses F from the l e f t , and (b) /[G(x)-F(x)]dx = 0. (2.12) In D e f i n i t i o n 2.8, condition (b) implies that the mean of G and F i s i d e n t i c a l ; condition (a) implies that G has a greater v a r i a b i l i t y than F. For a mean-variance type agent, F c l e a r l y dominates G. The single cross- ing requirement i s however not t r a n s i t i v e . To see t h i s , suppose F, G and H are three d i s t r i b u t i o n s with the same mean. That H single-crosses G and G single-crosses F from the l e f t does not imply that H w i l l single-cross F. The mean preserving spread defined below v i a second-degree stochastic dominance i s less r e s t r i c t i v e but t r a n s i t i v e . D e f i n i t i o n 2.9: For F, G e Dj, F i s said to s t o c h a s t i c a l l y dominate G in the second degree, denoted by F >z G, i f T(y) = lj„ [G(x)-F(x)]dx > 0 for a l l y e J , (2.13) and T(«) = / J[G(x)-F(x)] = £ JG ( x ) - F ( x ) ] d x = 0. (2.14) A l t e r n a t i v e l y , G i s said to be a mean preserving spread (mps) of F. When the means of F and G e x i s t , condition (2.14) implies that they are equal. Condition (2.13), i n contrast, represents a requirement on th e i r 'squeezed' means — i f we a r b i t r a r i l y pick a point y and concentrate a l l the mass over [y, m) onto y, then the squeezed mean of G must not be greater than that of F. This can be seen by rewriting condition (2.13) as (2.15) below: T(y) = |/JxdF(x)+y[l-F(y)] }-{/JxdG(x)+y[l-G(y)] } > 0 for a l l y. (2.15) Condition (2.13) w i l l obtain i f F and G have the single crossing pro- - 41 - perty and s a t i s f y the equal mean condi t ion (2 .14) . Simple mps i s there- fore a s p e c i a l case of mps. In f a c t , Rothschi ld and S t i g l i t z (1970) show that an mps of F can be viewed as a r e s u l t of a sequence of simple mps' of F . D e f i n i t i o n 2.10; A dec i s ion maker's preference i s said to d i sp lay mps r i s k avers ion (MRA) at F i f he always prefers F to G whenever F > 2 G. His preference i s said to d i s p l a y g lobal MRA (GMRA) i f i t d i sp lays MRA at a l l F . Lemma 2 . 1 : If F > 2 G and F t t = (l-oc)F+aG, then F > 2 F t t >2 F " ' >2 G for any a, a' e (0,1) such that a < a ' . Hence, regardless of the underly ing preference theory, Axiom 4 (Mono- t o n i c i t y ) impl ies that V ( F a ) must decrease i n a i f V i s to d i sp lay GMRA. Theorem U5.1 (GMRA): For F , G e D , F a = (l-a)F+aG where a e (0 ,1 ) , and «J any preference f u n c t i o n a l V: D ->• R s a t i s f y i n g Assumption 1.1, F > 2 G implies V(F) > V(G) i f and only i f F > 2 G implies ^ V(F t t ) < 0. Another way of c h a r a c t e r i z i n g r i s k aversion i n the large i s v i a c e r - t a i n t y equiva lent . D e f i n i t i o n 2.11: If a dec i s i on maker is i n d i f f e r e n t between F and 6 , then c c i s said to be h is (uncondi t ional ) c e r t a i n t y equivalent (CE) of F . D e f i n i t i o n 2.12: If a dec i s ion maker i s i n d i f f e r e n t between two compound l o t t e r i e s pF+(l-p)H and p6 c +( l -p )H, then c i s said to be his c o n d i t i o n a l c e r t a i n t y equivalent (CCE) of F , c o n d i t i o n a l on p r o b a b i l i t y p and d i s - t r i b u t i o n H. - 42 - D e f i n i t i o n 2 » 1 3 : I f a dec i s i on maker always prefers ° y X ( j p t 0 F> then his preference i s sa id to d i sp lay (uncondit ional ) c e r t a i n t y equivalent r i s k avers ion (CERA) at F . His preference i s sa id to d i sp lay g loba l CERA (GCERA) i f i t d i sp lays CERA at a l l F . D e f i n i t i o n 2 .14: For any F , H E D J and p e ( 0 , 1 ] , i f a dec i s ion maker always p r e f e r s (1 -p) ^y X ( jp + PH to ( l -p)F+pH, then his preference i s said to d i sp lay c o n d i t i o n a l c e r t a i n t y equivalent r i s k avers ion (CCERA) at F . His preference i s sa id to d i sp lay g loba l CCERA (GCCERA) i f i t d i sp lays CCERA at a l l F . D e f i n i t i o n 2.13 (2.14) impl ies that i f a dec i s ion maker i s GCERA (GCCERA), then his CE (CCE) of any l o t t e r y is always smaller than the expected value of that l o t t e r y . Under expected u t i l i t y , the s u b s t i t u t i o n axiom requires that i f G ~ F , then for any d i s t r i b u t i o n H and p r o b a b i l i t y p , i t must be true that pG+(l-p)H ~ pF+( l -p)H. Since 6 c ~ F , the s u b s t i - t u t i o n axiom implies that the CE and CCE of any d i s t r i b u t i o n are i d e n t i - c a l . Therefore , CERA and CCERA are equivalent under expected u t i l i t y . Beyond expected u t i l i t y , CCERA i s weaker than the s u b s t i t u t i o n axiom. I t simply requires that i f p6 c +( l -p)H ~ pF+(l -p)H, then c < /xdF for a l l p and H. As such, an agent's CE and CCE of a d i s t r i b u t i o n F need not be equal . A dec i s ion maker's CE can be in terpreted as the amount he must be paid to give up a l o t t e r y with p o s i t i v e expected va lue . The insurance premium given i n D e f i n i t i o n 2.3 i s a form of CE s ince , in an insurance context , agents can be viewed as seeking to s e l l adverse r i s k s — Given that he has been endowed with a r i s k of negative expected value , how much - 43 - would he be w i l l i n g to pay for trading away this risk? Therefore, we can also use insurance premia to characterize global r i s k aversion as below (IPRA and CIPRA are defined i n D e f i n i t i o n s 2.3 and 2.4, r e s p e c t i v e l y ) : D e f i n i t i o n 2.15: A decision maker's preference i s said to display global (unconditional) IPRA (GIPRA) i f i t displays IPRA at a l l wealth l e v e l s x. D e f i n i t i o n 2.16: A decision maker's preference i s said to display global c o n d i t i o n a l IPRA (GCIPRA) i f i t displays CIPRA at a l l wealth l e v e l s x. As CERA and IPRA are equivalent (so are CCERA and CIPRA), we w i l l draw only the one more relevant to the issue under discussion. Now that we have introduced GCCERA, i t i s necessary to impose lower boundedness on J . Consider the following property of >: D e f i n i t i o n 2.17: A preference ordering > i s said to display continuity i n d i s t r i b u t i o n (CD) i f whenever F >- G and the sequence {G^} converges to G i n d i s t r i b u t i o n , there exists an N > 0 such that for every n > N, F >- V Suppose f u n c t i o n a l V represents preference ordering >. When G^, n = 0, 1 have compactsupports, CD means that |V(G n)} n™ 0 w i l l converge to V(G) i f \G \ ™ converges to G. G r a p h i c a l l y , CD means that the 'not-1 n Jn=0 worse-than set' i s closed, or the 'better-than set' i s open. C l e a r l y , CD implies Axiom 3 ( S o l v a b i l i t y ) . In order for GCCERA and CD to be compatible, J must bounded from below. To i l l u s t r a t e , consider F^ = (1-q) 6 x +g+q& x_j Q/q)_]_ j Q> where 9 > 0, q e (0,11. SD and GCCERA imply 6 ,. >- 6 > F for q e (0,0.5], but F » H \ > J f J X + Q x ~ q q converges i n d i s t r i b u t i o n to 6 x + g as q -> 0, contradicting CD. In the re- - 44 - mainder of this essay, we w i l l assume that J i s bounded from below when- ever GCCERA. i s invo lved . So f a r , we have given four d e f i n i t i o n s of uncondi t ional g loba l r i s k avers ion , i . e . PLRA, GMRA, GCERA and GIPRA, and two of c o n d i t i o n a l ones, i . e . GCCERA and GCIPRA. The fo l lowing i s obvious i n l i g h t of t h e i r d e f i n i t i o n s : C o r o l l a r y 2.3: GCCERA + GCERA -»• PLRA. We have also pointed out that GCERA and GIPRA are equivalent (so are GCCERA and GCIPRA). Beyond t h i s , how are they l inked together? It turns out that , GCCERA and GMRA are equivalent regardless of the under ly ing preference theor i e s . In what fo l lows , we f i r s t prove th i s equivalence for 'elementary l o t t e r i e s ' and then extend i t to a r b i t r a r y monetary l o t t e r i e s . D e f i n i t i o n 2.18: A l o t t e r y of the form s A i o = -̂-6 + . . . + -U i s i= l N x_̂ N ^ c a l l e d an elementary l o t t e r y , denoted by x = ( x ^ , x 2 , • • • , x ^ ) . In other words, an elementary l o t t e r y i s a l o t t e r y which gives N out- comes x^ , x^ w i t h u n i f o r m p r o b a b i l i t y 1/N. Note that x^, x^ need not be d i s t i n c t . Thus, any l o t t e r y invo lv ing a f i n i t e number of outcomes with r a t i o n a l p r o b a b i l i t i e s can be expressed as an elementary l o t t e r y . The fo l lowing i s due to Hardy, Li t t lewood and Poyla (1934): N D e f i n i t i o n 2 .19 : For v e c t o r s x , £ e J , x i s a major iza t ion of £ (or x majorizes y j , denoted by x >m £ , i f (a) 2 . n , x . > E . n , y , f ° r a 1 1 1 < n < N (2.16) v ' i= l i i = l - , i and (b) E i f 1 x j L = E ^ y . , (2.17) where the elements of x and y_ have been arranged i n ascending order . When i n e q u a l i t y (2 .16) holds s t r i c t l y for at l east one n, x i s sa id to majorize £ s t r i c t l y . In D e f i n i t i o n 2.19, condi t ion (b) impl ies that x and £ share the same mean. There i s a sense i n condi t ion (a) that x i s more 'centered' towards the mean than £ when x majorizes £ . This sounds s i m i l a r to the mean pre- serving spreads given i n D e f i n i t i o n 2 .9 . As i t turns out, major iza t ion i s equivalent to the second-degree s tochas t i c dominance for elementary l o t t e r i e s . Lemma 2.2: For elementary l o t t e r i e s , x >m £ i f and only i f x > 2 £ . P r o o f : E x p r e s s x and y as F = E . N , -JH5 and G = S . N . -̂ -6 , r e s p e c t i v e l y . ~ ^ 1=1 N x. i= l N y ' v J l J i Since = \ = L N X i = N Ei=lXi' J I J HIGCX ) = E . = 1 - y ± = - E i = i y i , condi t ion (2.14) impl ies and i s impl ied by equa l i ty (2 .17) . (Su f f i c i ency ) We prove (2.16) by induct ion as fo l lows . F i r s t , we show that x^ > y^. Suppose the contrary that x^ < y^. Without loss of gene- r a l i t y , a l s o assume x < y, < x , , , where n = 1, or N. Consider z J ' n J 1 n+1' ' ' = y^. Then, fz , „ , > . r i r i / M 1 ~ n . N—n 1 , „ n . N—n J ^ x d F ^ ) + z [ l - F ( z ) ] = _ E i = 1 x . + _ z = ^ X ; L + E . = 2 x . + — z . 1 , n-1 , N-n < N x i + - r z + - r z < 1 y l + T z = ^ - « x d G ( x ) + Z [ 1 " G ( Z ) ] - This contrad ic t s (2 .15) , a condi t ion for F > 2 G. Therefore , x^ > y^. - 46 - We next assume that, for some k < N, E i = i x i > E i = i y r ( 2 a 8 ) k+l k+1 It remains to be shown that ^ _ ^ x ^ > ^ i = l y i " S u P P o s e the contrary that ^ x ± < ^ y ± - (2.19) I n e q u a l i t i e s (2.18) and (2.19) together imply that < yk+l* A s s u m e without loss of generality that x ^ + ^ < x n < y k + i * Xn+1* L e t Z = yk+l° J!,xdF(x) + z [ l - F ( z ) ] - I S i; ix ± + ^ z 1 k+l 1 n N-n N 2 j i = l X i N i=k+2 Xi N Z . 1 vk+l . N-k-1 < — E. ,X, + Z N 1=1 i N , 1 Jc+1 , N-k-1 < N S i = l y i + "IT" Z = £,xdG(x) + z [ l - G ( z ) ] . k+l Jc+1 T h i s again c o n t r a d i c t s (2.15). Hence, ^_^x^ > ^_^y^* By induction, i t follows that E . n i X j > E ^ y . for a l l 1 < n < N. (2.16) i=l i i = l l ( N e c e s s i t y ) For any z e J, suppose without loss of generality that XR < z < x , and y, < z < y, ., . There are three cases to consider: ( i ) k = n+1 k ^k+1 n, ( i i ) k < n, ( i i i ) k > n. Case ( i ) : k = n J!„*dF(x) + = ± \l^± + ^ ' . 1 „ n , N-n " N z i = i y i + ~ir z = J_ oxdG(x) + z[l-G ( z ) ] Case ( i i ) : k < n Jf^dFCx) + z[l-F ( z ) ] = I E . ^ x 1 + 2 = » z - 47 - , 1 „ n , N-n > N S i = l y i + T " z 1 „ k , l „ n , N-n = N E i = l y i + N Z i = k + l y i + T Z . 1 „ k , N-k > N S i = l y i + — Z = jf^xdGCx) + z [ l - G ( z ) ] . Case ( i i i ) : k > n J f J«1F<X) + z [ l - F ( z ) ] = 1 E i ^ x 1 + 3jp z . 1 „ n k-n N-k > N s i = i y i + i r z + - r z . 1 „ n . 1 „ k N-k N i= l i N i=n+lJx N 1 _ k N-k = N E i = i y i + - r z = jf^xdGCx) + z [ l - G ( z ) ] . Q.E.D. We proved Lemma 2.2 by v e r i f y i n g that the condit ions for major iza t ion are equivalent to the condit ions for second-degree s tochast ic dominance. I t i s easy to check that condi t ion (2.14) i s s a t i s f i e d for elementary l o t t e r i e s x and £ i f and on ly i f (2.17) holds . The equivalence between (2.15) and (2.16) i s not as d i r e c t . That (2.16) implies (2.15) i s v e r i - f i ed v i a s tra ightforward a lgebra . That (2.16) i s also necessary for (2.15) i s proved by i n d u c t i o n . For elementary l o t t e r i e s , condi t ion (2.15) impl ies that the 'z-squee- zed mean' of x ( i . e . , the p r o b a b i l i t y measure on [z,x^] i s squeezed to the point z , where z < x^) must not be smaller than the l ikewise squeezed mean of In c o n t r a s t , c o n d i t i o n (2 .16) says that the 'n-element p a r t i a l mean' of x ( i . e . , the mean of a reduced vector ( x ^ , . . . , x n ) with n < N) - 48 - must not be l e s s than that of Note that a 'squeezed l o t t e r y ' i s the o r i g i n a l l o t t e r y with the r i g h t t a i l beyond a f ixed point being 'squeezed' to that point while a ' p a r t i a l l o t t e r y ' i s a truncated l o t t e r y of the o r i - g i n a l one with uniform (cond i t i ona l ) p r o b a b i l i t y 1/n. The fo l lowing ob- servat ion should provide more i n t u i t i o n for the equivalence between (2.15) and (2.16) . C o r o l l a r y 2.4: For elementary l o t t e r i e s x, £ e , x >m £ impl ies jn i jn ,m k jn jn N - l _ x > x. . . . > £ > . . . > £ = y_> where X - ( X 1 } X 2 , X 3 , . . . , X n , X n + 1 , X n + 2 , . . . . X ^ . X j j ) Zl " ( y i ' x 2 + ( x r y l ) ' x 3 ' " , , X n ' X n + l , X n + 2 ' , , , , X N - l , X N ) n z = (yi.y2«y3' • ' yn-V i + E i - i ( V i ) • V 2 » • • • • V i 'V (2*20) x,N_1 = ( y 1 . y 2 » y 3 ' - - - ' y n ' y r r i - i ' y n + i ' - - - ' % - i ' x N + C i ( x r y i ) ) = ( y i . y 2 ' y 3 ' - - - ' y n ' y n + i ' y n + i ' - - - » y N - i ' y N ) =*>' Proof: Omitted s ince i t i s s t ra ight forward . C o r o l l a r y 2.4 means t h a t , i f x majorizes then £ can be obtained from x v i a a f i n i t e sequence of mps' or major iza t ions . From (2.16) and (2 .17) , we know that x^ > y^ and < . S t a r t i n g from x, y^1 i s ob ta in - ed by pushing x^ leftwards to y^ and simultaneously pushing x 2 rightwards by a distance of x ^ - y ^ . Label th i s new p o s i t i o n as z 2 . To obtain £ 2 from y^1, a g a i n push z 2 l e f twards to y 2 (by a distance of [x 2 +(x^-y^)]-y 2 ) and push X g r ightwards by the same distance to a p o s i t i o n labeled z^. C o n t i - - 49 - nue t h i s process u n t i l £ r e s u l t s . Corollary 2.4 t e l l s us that £ w i l l be obtained after N-l such operations. Since at the i t h step ( i = 1, 2, ..., or N - l ) , we .push x^ downward and x ^ + ^ upward by the same distance, yj1 must be an mps of yj 1 . x, >m implies that, after each i t e r a t i o n , say the i t h one, z. , (the p o s i t i o n where x. , has been pushed to) must be to the ' l + l r l + l r i g h t of y. , so that at the next i t e r a t i o n , the push of z.,, to y... i s 6 J i+l l+l •'l+l always a leftward one. Only N-l, rather than N, i t e r a t i o n s are needed be- cause z„ must coincide with y„ i f x and y are to have the same mean. N N To show that GCCERA and GMRA are equivalent for elementary l o t t e r i e s , we need Lemma 2.3 below: Lemma 2.3: Under completeness, t r a n s i t i v i t y and SD, GCCERA implies that, for every a, e, 9, p ( e, 9 > 0, p e (0,1]), and H e Dj, F E*4 6a-e +TW + ( 1 " P ) H >~ *46a-e-9 + -k+s+9> + ( 1 ^ ) H = G " Proof: Let q = e/(e+9). Then, F > p f 5 a - £ - e + i ? 6 a + i W + (1-P)H > P r|Vs-0+ ( 1 - ^ 6 a + l 6 a + e + 9 l + (l-p)H > G. Q.E.D. Theorem U5.2 (GRA) : For elementary l o t t e r i e s and preference ordering > s a t i s f y i n g completeness, t r a n s i t i v i t y and SD, GCCERA <=> GMRA. Proof: (<=) This is straightforward. (=>) Suppose x = £. N — 6 >2 £. N, —6 = y. Lemma 2.2 t e l l s us that x ~ i=l N x. i=l N y. ~ l J i >m y,, which by C o r o l l a r y 2.4 implies that £ can be obtained from x v i a - 50 - the sequence yj 1 g i v e n by ( 2 . 2 0 ) . Since yj 1 ^ and yj 1 are the fo l lowing elementary l o t t e r i e s : n-1 _ y N I t N-2 i 1 , y N l o i *• " i= l N n-1 N 1 i= l N ^ V T-n+2 N ^ x , ] y J l i + I ^ 5 x + E n _ 1 ( x -y )+ I 6 x J ^ V > l l X i y i ; n+l n _ N 1 . _ N-2 r _ n - l 1 . N 1 * 1 X. = E i = l N n " T i E i = l l F 2 6 y , + Ei=n+2 N=2 6 x . < y± i i + l { K + I 6 x J . 1 + S , n 1 ( x . - y . ) J » y n n+l i= l l J i Lemma 2.3 impl ies that . 1 v v n v v N-1 _ C ••• <,X< <t * * * X. = X,' Q . E . D . The next task i s to extend Theorem U5.2 from elementary l o t t e r i e s to general monetary l o t t e r i e s . This can be done v i a CD. Theorem U5.3 (GRA): Under completeness, t r a n s i t i v i t y , SD and CD, GCCERA <=> GMRA. Proof: Omitted since i t i s a s p e c i a l case of Theorem U7.2 . Theorems U5.1 - U5.3 are r e s u l t s on GRA. Because they only require preferences to be complete, t r a n s i t i v e , consistent with SD (and i n a d d i - t i o n be continuous in d i s t r i b u t i o n for Theorem U5.3) , and do not depend on s p e c i f i c preference func t iona l forms, we regard them as ' t h e o r y - f r e e ' , and according ly l a b e l them with l e t t e r U. In the l i t e r a t u r e of r i s k avers ion , i t i s wel l known that , under EU, GMRA <=> GCCERA <=> GCERA <=> concavity of u . Machina (1982a) proves that , for the more general Fre*chet d i f f e r e n t i a b l e u t i l i t y , GMRA <=> GCCERA <=> the concavity of l o c a l u t i l i t y functions u ( x ; F ) . Theorem U5.3 t e l l s us that the equivalence between GMRA and GCCERA i s a c t u a l l y more fundamen- t a l than bel ieved and does not even depend on Fre"chet d i f f e r e n t i a b i l i t y . I t i s true for EU, WU, LIU, FDU, as wel l as LGU. When more s tructures are imposed on the preference f u n c t i o n a l , s t r o n - ger impl i ca t ions of g loba l r i s k avers ion are obta inable . The fo l lowing theorem on EU i s we l l known. Theorem EU5 (GRA): For an EU dec i s ion maker with a continuous, increas ing von Neumann-Morgenstern u t i l i t y funct ion u(x) , the fo l lowing propert ies are equivalent: (a) GCCERA; (b) (Concavity) u(x) i s concave; (c) GCERA; (d) PLRA. Given Theorem EU5, we may say that an EU dec i s ion maker i s GRA i f he has a concave u t i l i t y funct ion , and i s consequently GCCERA, GMRA, GCERA and PLRA. Once we depart from EU, we must be s p e c i f i c about the sense of g l o b a l r i s k avers ion being re ferred to . For instance , under WU which weakens the (strong) s u b s t i t u t i o n axiom to the weak s u b s t i t u t i o n axiom, a c o n d i t i o n a l r i s k avers ion d e f i n i t i o n w i l l imply, but w i l l not be equiva- lent to , i t s uncondi t iona l counterpart . Theorem WU5 (GRA): For a WU dec i s ion maker (v,w) with LOSUF C ( x ; F ) , the fo l lowing are equivalent: (a) GCCERA; (b) (Concavity) C(x;F) i s concave i n x for a l l F; Proof: (a) -+ (b): Suppose there ex i s t s H such that C(x;H) i s s t r i c t l y con- vex i n x. Then, for any x^ < x 2 < x^, there ex is t s q such that - 52 - C(x ;H)-C(x ;H) ° < C(X3;H)-C(XI ;H) < « < 2 ' 2 2 > X 2 ~ X 1 q < _ < 1. (2.23) X 3 X l Inequality (2.22) implies that C(x2;H) < qC(x 3;H)+(l-q)C(x 1;H). (2.24) Define F = q6 +(l-q)6 . (2.24) becomes x3 X l /C(x;H)d6 < /C(x;H)dF, X2 or /C(x;H)d[F-6 v ] > 0. (2.25) 2 Let G = (l-p)F+pH and G' = (1-p)6 +pH. Since by expression (1.9), X2 3p WU(G)| p = 1 = /C(x;H)d[H-F] and d WU(G')| = /C(x;H)d[H-6 x ], dp lp=. - 2 inequality (2.25) implies d dp /C(x;H)d[F-6 x ] = ?L [WU(G')-WU(G)]| p = 1 > 0. Since WU(G')j p = 1 = WU(H) = WU(G)| p = 1, we have WU((l-p)6 +pH) < WU((l-p)F+pH) (2.26) X2 f o r some p s u f f i c i e n t l y c l o s e to 1. Since x^ > qx^+(l-q)x^ = JxdF(x) from (2.23), stochastic dominance implies that WU((1-p)6x^+pH) > WU((1-p)6 / x d F+pH)• (2.27) (2.26) and (2.27) together imply WU((l-p)F+pH) > WU((l-p)6 / x d p+pH), con- t r a d i c t i n g GCCERA. (b) -> ( a ) : Let F , G e D be such that F > z G and define F = ( l -a)F+aG, vJ a e ( 0 , 1 ) . Extend expression (1.24) as fo l lows: = -/C ' (x ;F a ) [G(x) -F(x) ]dx (1.24) = -|C ' ( x ;F a )d /_ X 3 [G( t ) -F ( t ) ]d t = -JC ' (x ;F a )dT(x) (from (2.13)) = jT(x)dC ' (x ;F a ) - /T(x)C"(x;F a )dx . (2.28) Given T(x) > 0 for a l l x, C(x;F) being concave i n x for a l l F im- dWU(F a) p l i e s that — ^ — - < 0. By Theorem U5.1 , t h i s y i e l d s GMRA. Since GMRA <=> GCCERA according to Theorem U5.3, we have GCCERA. Q . E . D . Theorems WU5, U5.3 and C o r o l l a r y 2.3 together give the fo l lowing r e l a t i o n s for WU: C"(x;F) < 0 at a l l F <=> GMRA <=> GCCERA => GCERA => PLRA. The fo l lowing example i s provided to demonstrate that , under WU, PLRA does not imply GCCERA i n genera l . Example 2 .1: (PLRA does not imply GCCERA under WU.) PLRA requires r ( x ) = - [ I ^ + ^ l > 0 (2.29) L v ' (x ) w(x) 1 f o r a l l x . P i c k v ( x ) = a+bx, b > 0, x e [ 0, 0 0 ) . We have v"(x) = 0. Consequently, r (x) = - ^ ^ x ^ ^ 0 for a l l x, which implies w'(x) < 0 for a l l x. (Obviously , we are not in teres ted i n the case with w'(x) = 0 . ) Given v"(x) = 0 for a l l x, the GCCERA condi t ion reduces to r ( x ; F ) = 2w'(x)v'(x) + w-(x)[v(x)-WU(F)] % ( 2 > 3 0 ) In (2 .30) , the f i r s t term i n the numerator of the RHS i s always nega- - 54 - t i v e . For a s t r i c t l y convex, decreasing w, we can always construct a d i s t r i b u t i o n F with WU(F) s u f f i c i e n t l y small so that the second term w"(x)[v(x)-WU(F)] i s s u f f i c i e n t l y p o s i t i v e , leading to C"(x;F) > 0. S i m i l a r l y , for a s t r i c t l y concave, decreasing w, we can always construct a d i s t r i b u t i o n F with WU(F) s u f f i c i e n t l y large so that C"(x;F) > 0. Hence, PLRA does not imply CCERA i n genera l . I Two questions are of i n t e r e s t here: (1) Under what condit ions w i l l PLRA imply GCCERA? (2) When w i l l PLRA imply GCERA? The condi t ion for PLRA i s r ( x ) = _ t y ^ +IwHxij > 0 ( 2 . 2 9 ) v' (x) w(x) at a l l x. The cond i t ion for GCCERA i s C ( x ; F ) = w"(x)[v(x)-WU(F)]+2w'(x)v'(x)+w(x)v"(x) < Q (2.31) ' JwdF for a l l F , which i m p l i e s , a f t er omitt ing the argument x: v—v — ! v" , 2w' I . I W" W V-V i /o oo\ - L —r + J > m a x t r » ~ r }» (2.32) V w w v' w v' ' which can be restated as condit ions (2.33) and (2.34) below, noting w, v' > 0: - | _ ^ + — J > — — ^ i f w" > 0, (2.33) v w w V I V" , 2w' I . W" V-V „ . n /o o/N - — r + > r i f w < 0. (2.34) V W W V The LHS of (2.33) and (2.34) i s the WU Arrow-Pratt index. PLRA and GCCERA w i l l be equivalent i f condi t ion (2.29) and condi t ion (2.33) or (2.34) co- i n c i d e . C l e a r l y , a l i n e a r weight funct ion w i l l do. C o r o l l a r y 2 .5: For a WU dec i s ion maker (v,w) with l i n e a r weight funct ion w, the fo l lowing are equiva lent : (a) GCCERA; (b) C(x;F) i s concave i n x for a l l F; (c) GCERA; (d) PLRA. Proof: Omitted. According to th is c o r o l l a r y , a dec i s ion maker's preference can d i s - play GMRA, GCCERA, GCERA and PLRA simultaneously, and yet does not subs- c r i b e to EU. This suggests a p o t e n t i a l choice of preference model to those who recognize the r e s t r i c t i v e n e s s of EU but are appal led at the complexity of an ' a l l - o u t ' WU. We now turn to the second quest ion: When w i l l PLRA and GCERA be equ i - valent? According to C o r o l l a r y 2 .5 , they are equivalent when w i s l i n e a r . This i s only s u f f i c i e n t however. There are other condit ions under which PLRA w i l l imply GCERA (but not n e c e s s a r i l y GCCERA). For instance , c o n s i - der a concave v . A s u f f i c i e n t condi t ion for PLRA i s w decreas ing . R e c a l l that OT<r> - /V/wjx)dF(xSX) = Mx)dFW(x), (1.19) where / X o w ( t ) d F ( t ) £ \ ( t ) d F ( t ) I f w i s decreasing, F w i l l s t o c h a s t i c a l l y dominate F w i n the f i r s t degree. Lemma 2.4: Suppose the weight funct ion w i s decreas ing. Then, for any d i s t r i b u t i o n F , F > i F W . Proof: Omitted. Since (v,w) with v concave w i l l d i sp lay GCERA when w i s constant (a case of EU) , (v,w) w i l l be even more i n c l i n e d to d i sp lay GCERA when w i s s t r i c t l y decreas ing . In such a case, a l i n e a r v w i l l su f f i ce for GCERA. - 56 - F w ( x ) = — . (1.20) Given the d i scuss ion above, C o r o l l a r y 2.6 i s stated without proof . C o r o l l a r y 2.6: If w i s decreasing and v i s concave and at l east one of the condi t ions holds s t r i c t l y , then PLRA i s equivalent to GCERA. The r e s u l t of Theorem WU5 can be extended to the more general LGU: Theorem LGU5 (GRA): For an LGU funct iona l V: Dj -»• R with LOSUF £: J * Dj •* R, the fo l lowing are equivalent: (a) GCCERA; (b) (Concavity) C(x;F) i s concave i n x for a l l F . Proof: Omitted since i t i s s i m i l a r to the proof of Theorem WU5. Comparing Theorems EU5, WU5 and LGU5, we observe the fo l lowing s i m i - l a r i t i e s and d i s t i n c t i o n s . F i r s t , the von Neumann-Morgenstern u t i l i t y funct ion u(x) i n condi t ion (b) of Theorem EU5 i s replaced by the LOSUF C(x;F) i n both Theorem WU5 and Theorem LGU5, confirming that the LOSUF C(x;F) i s the non-EU equivalent of the von Neumann-Morgenstern u t i l i t y f u n c t i o n . Second, the GCCERA condi t ion appears in a l l theorems because i t i s the strongest form of g loba l r i s k avers ion i n the sense that i t impl ies both GCERA and PLRA. T h i r d , un l ike Theorem EU5, the GCERA and PLRA cond i - t ions are absent i n Theorems WU5 and LGU5 because they are implied by, but not equivalent to , the GCCERA and concavity c o n d i t i o n s . In the next s ec t ion , we w i l l u t i l i z e an agent's demand for r i s k y asset to introduce another d e f i n i t i o n of g loba l r i s k avers ion , c a l l e d ' p o r t f o l i o r i s k a v e r s i o n ' . - 57 - 3 PORTFOLIO CHOICE PROBLEM From a finance viewpoint , our ul t imate in t ere s t i n r i s k avers ion l i e s i n i t s impl ica t ions for asset demand. We s h a l l present the r e s u l t in th is d i r e c t i o n under expected u t i l i t y and see how other preference func t iona l approaches depart from i t . We r e s t r i c t our i n v e s t i g a t i o n to a one-safe- asse t -one-r i sky-asse t world . D e f i n i t i o n 3 .1 : An investment environment which provides only one safe a s s e t w i th ( g r o s s ) r a t e of r e t u r n r , and one r i s k y asset with (gross) rate of re turn z i s c a l l e d a simple p o r t f o l i o se t -up . We w i l l hereafter re fer to an asset by i t s rate of r e t u r n . The nota- t ions to be used in th i s sec t ion are summarized below: r : gross rate of re turn on the safe asset; z: gross rate of re turn on the r i s k y asset; y Q : p o s i t i v e i n i t i a l wealth; x: d o l l a r amount invested i n the r i s k y asset; y -x : d o l l a r amount invested i n the safe asset; 6: proport ion of y Q invested i n the r i s k y asset; 1-8: proport ion of y Q invested i n the safe asset; y: f i n a l wealth. D e f i n i t i o n 3 .2: Problem (3.1) below i s an i n v e s t o r ' s (uncondi t ional ) simple p o r t f o l i o choice (SPC) problem: To f ind x* such that , for every x # x*, F ~ * > F ~ , (3.1) where - 58 - y = y Q r + x ( z - r ) (3.2) y* = y Q r + x*(z -r ) (3.3) D e f i n i t i o n 3 .3: Problem (3.4) below i s an inves tor ' s c o n d i t i o n a l simple p o r t f o l i o choice (CSPC) problem: To f ind x* such that , for every x * x*, P F~*+( l -p )H > p F M - ( l - p ) H , (3.4) where y and y* are g i v e n by (3 .2) and (3 .3 ) , r e s p e c t i v e l y , p e ( 0 ,1 ] , and H i s a d i s t r i b u t i o n independent of F . Without a p r i o r i r e s t r i c t i o n s on the preference order ing , i t i s not guaranteed that the optimal x* w i l l be unique. I t seems des irab le to im- pose the fo l lowing r e g u l a r i t y : D e f i n i t i o n 3.4: In a simple p o r t f o l i o set-up with safe asset r and r i s k y a s se t z , an i n v e s t o r with i n i t i a l wealth y i s said to be an (uncondi- o - — t i o n a l ) d i v e r s i f i e r i f his preferences over the set of d i s t r i b u t i o n s {F / ~ _ \} are s t r i c t l y quasi-concave i n x. y o X D e f i n i t i o n 3 .5: In a simple p o r t f o l i o set-up with safe asset r and r i s k y a s s e t z , an investor with i n i t i a l wealth y i s sa id to be a c o n d i t i o n a l » J Q d i v e r s i f i e r , c o n d i t i o n a l on p and H, i f his preferences over the set of d i s t r i b u t i o n s {pF r + x (~_ \+(l-p)H} are s t r i c t l y quasi-concave i n x. y o Z Again , l e t us use a simplex of 3-outcome l o t t e r i e s to i l l u s t r a t e the i n t e r p r e t a t i o n of an investor being a d i v e r s i f i e r . R e c a l l that , in such a simplex, the ind i f f erence curves are p a r a l l e l s t ra ight l ine s for EU-type preferences , are n o n p a r a l l e l s t ra ight l i n e s fanning out from an ex ter ior point for WU-type preferences , and are a r b i t r a r y nonintersect ing smooth - 59 - curves f o r LGU-type preferences . Assume F ~ = (1-p) 6 +p6- and 6 , 6 , jf t- z z r z v z ' v r ' o- o 6 - a r e the three v e r t i c e s of a simplex, where z < r < z and pz+(l-p)z > y o Z r . When x = 0, F ~ = 6 . As x increases , F ~ w i l l move along a path and y y 0 r y V ( F ~ ) w i l l v a r y . At x = y , the path reaches (1-p) 6 +p6 - . In gene- y o yQZ yQZ r a l , th i s path w i l l cross numerous ind i f f erence curves . If V i s s t r i c t l y q u a s i - c o n c a v e i n x, V(F~) w i l l e i ther be monotone from x = 0 to x = y Q or increase i n x up to the optimal x* and then decrease. This w i l l be ensured i f a l l be t ter - than sets i n the simplex are convex. C l e a r l y , a GMRA EU or WU investor i s g e n e r i c a l l y an uncond i t i ona l , as wel l as c o n d i - t i o n a l , d i v e r s i f e r . C o r o l l a r y 3 .1: A s t r i c t l y GMRA WU investor i s always a c o n d i t i o n a l d i v e r - s i f i e r . Proof: The CSPC problem for a WU investor (v,w) with LOSUF C i s Max WU(pF r + x f ~ - r ) + ( 1 ~ p ) H ) - ( 3 - 5 ) x y o The f i r s t and second d e r i v a t i v e s of WU above w . r . t . x are given i n (3.6) and (3.7) below, r e s p e c t i v e l y : J C ( y ; G ) ( z - r ) d F ( z ) , (3.6) /C " ( y ; G ) ( z - r ) 2 d F ( z ) , (3.7) where G = p F ^ * f ( l - p ) H and y = y Q r + x ( z - r ) . With C(y;G) s t r i c t l y concave i n y for a l l G e D j , (3 .7 ) i s always n e g a t i v e , i . e . WU i s s t r i c t l y quasi-concave i n x. Q . E . D . On the contrary , an LGU func t iona l V(F) does not general ly possess such a property . For example, Dekel (1984) showed that , under FDU, the - 60 - concavity of l o c a l u t i l i t y functions u(x;F) and the quasi-concavity of the preference functional V(F) are j o i n t l y s u f f i c i e n t for the demand prefer- ences over assets to be quasi-concave. For LGU, the requirement of an i n - vestor being a d i v e r s i f e r means that some degree of a r b i t r a r i n e s s i n the in d i f f e r e n c e curves i s removed so as to rule out cases where simple port- f o l i o choice decisions might y i e l d undesirable multiple solutions. We define the x* that solves an investor's SPC problem (3.1) or CSPC problem (3.4) as his demands for r i s k y asset z: D e f i n i t i o n 3.6: Suppose x* solves the SPC problem (3.1) uniquely for an investor with i n i t i a l wealth y . Then, x* i s c a l l e d his (unconditional) money r i s k y - a s s e t demand at y Q and 8* = x*/yQ i s c a l l e d his (uncondi- t i o n a l ) proportional risky-asset demand at y . D e f i n i t i o n 3.7: Suppose x* solves the CSPC problem (3.4) uniquely for an i n v e s t o r with i n i t i a l wealth y . Then, x* i s c a l l e d his condit i o n a l o money r i s k y - a s s e t demand at y Q and 8* = x*/y Q i s c a l l e d his condit i o n a l proportional risky-asset demand at y . When i t i s unambiguous, we may omit 'money' and 'proportional' i n the above terms. Note that we do not rule out shortsales i n both the SPC and the CSPC problems. This however w i l l not be an issue here because i t i s assumed throughout t h i s essay that E ( z ) > r , which implies that x* i s always nonnegative. Theorem U6 (Nonnegative Conditional Risky-Asset Demand): Suppose x* solves the CSPC problem (3.4) for any investor whose preferences are complete, t r a n s i t i v e and exhibit SD and GCCERA. Then, (1) x* (8*) > 0 i f E(z) > r; moreover, x* (8*) = 0 i f E(z) = r; (2) x* > 0 only i f E(z) > r. - 61 - Proof: (1) Suppose E(z) > r but x* < 0. Then, E(y*) = y Q r+x*[E(z ) -r ] < y Q r . SD impl ies that , for any p e [0,1) and H e D , a-p)S E (~*)+PH < (1-P)6 +pH. GCCERA impl ies ( ^ ^ ( y * ) ^ 1 1 ^ (1-P)F~*+PH. Hence, by t r a n s i t i v i t y , (1-p) 6 r+pH > ( l -p)F~*+pH. o This contradic t s the o p t i m a l i t y of x*. When E ( z ) = r , E (y ) = y Q r + x [ E ( z ) - r ] = y Q r for a l l x. GCCERA im- p l i e s ( l - p ^ ^ + p H = d - p ) 6 y r+pH > ( l - p ) FM- pH, implying x* = 0. (2) Suppose x* > 0 but E(z) < r . (We need not consider E(z) = r i n l i g h t of (1) a b o v e . ) T h e n , E ( y * ) = y Q r + x * [ E ( z ) - r ] < y Q r . GCCERA, SD and t r a n s i t i v i t y imply that ( l - p ) 6 y r+pH > ( l - p)6 E ( ~ ^ ) + p H > ( l -p )F~ A +pH, c o n t r a d i c t i n g op t ima l i ty of x* > 0. Q . E . D . The r e s u l t s of Theorem U6 can be out l ined as below: a. E(z) > r -> x* > 0; b . E(z) = r -> x* = 0; c . x* > 0 •*• E(z) > r . - 62 - Note that x* > 0 i s s u f f i c i e n t but not necessary for E(z) > r since x* > 0 i m p l i e s E ( z ) > r , which i n turn implies x* > 0. The equivalence however can be es tabl i shed under EU and WU. For an EU maximizer, the CSPC problem becomes: Maximize EU[pFH-(l -p)H] = p/u(y)dF~(y)+( l -p) Ju(s)dH(s) (3.8) x y y s . t . y = y^r + x ( z - r ) . The CSPC problem for a WU maximizer with value funct ion v and weight funct ion w i s : Maximize WU[pF~Kl-p)H] (3.5) x y where y = y Q r + x ( z - r ) . The opt imizat ion condit ions for (3.8) and (3.5) lead to the fo l lowing two theorems: Theorem EU6 (Pos i t i ve C o n d i t i o n a l Risky-Asset Demand): Suppose x* solves the CSPC problem (3.8) for an EU investor with an i n c r e a s i n g , s t r i c t l y concave u t i l i t y funct ion u (y ) . Then x* (8*) > 0 i f and only i f E(z) > r . Proof: Omitted since i t i s a s p e c i a l case of Theorem WU6. A l s o , see Arrow (1971). Theorem WU6 (Pos i t i ve Cond i t i ona l Risky-Asset Demand): Suppose x* solves the CSPC problem (3.5) for a WU investor (v,w) with i n c r e a s i n g , s t r i c t l y concave LOSUF C ( y ; F ) . Then, x* (8*) > 0 i f and only i f E(z) > r . Proof: In l i g h t of Theorem U6, we need to prove only the s u f f i c i e n c y . De- f ine G = p F ~ f ( l - p ) H . The FOC and SOC for x* to solve (3.5) are given by - 63 - by (3.9) and (3.10) below, r e s p e c t i v e l y : FOC: /C ' ( y* ;G) (z - r )dF(z ) = 0; (3.9) SOC: JC " ( y * ; G ) ( z - r ) 2 d F ( z ) < 0. (3.10) where C(y;G) - w(y)[v(y)-WU(G)]//wdG and y* = y Q r + x * ( z - r ) . Suppose E(z) > r but x* = 0. (Theorem U6 rules out the p o s s i b i l i t y of x* c o n t r a d i c t i n g FOC (3.9) and implying x* > 0. Q . E . D . Theorem WU6 t e l l s us that , l i k e h i s expected u t i l i t y counterpart ( c f . Theorem EU6), a GCCERA WU maximizer w i l l invest in the r i s k y asset i f and only i f the expected re turn on the r i s k y asset i s greater than the sure r e t u r n on the r i s k f r e e asset . Obvious ly , p o s i t i v e c o n d i t i o n a l r i s k y - a s s e t demand w i l l imply p o s i t i v e uncondi t iona l r i s k y - a s s e t demand. As a matter of f a c t , the l a t t e r requires only GCERA ( instead of GCCERA). The r e s u l t s i n Theorems U6, EU6 and WU6 are based on the assumption of GCCERA. I f an agent i s r i s k seeking, then i t i s quite natura l for him to have p o s i t i v e r i s k y - a s s e t demand when E(z) > r . This suggests yet ano- ther c h a r a c t e r i z a t i o n of r i s k avers ion: D e f i n i t i o n 3.8: In a simple p o r t f o l i o set-up with safe asset r and r i s k y asset z , the preference of an investor with i n i t i a l wealth y Q i s said to d i s p l a y ( u n c o n d i t i o n a l ) p o r t f o l i o r i s k a v e r s i o n (PRA) at y Q i f his r i s k y - a s s e t demand at y Q i s p o s i t i v e only i f E(z) > r . His preference i s sa id to d i sp lay g l o b a l PRA (GPRA) i f i t d i sp lays PRA at a l l y . < 0.) Then, /C ' ( y * ; G ) ( z - r ) d F ( z ) = C ' ( y o r ; G ) [ E ( z ) - r ] > 0, - 64 - D e f i n i t i o n 3 .9: In a simple p o r t f o l i o set-up with safe asset r and r i s k y asset z, the preference of an investor with i n i t i a l wealth y Q i s said to d i s p l a y c o n d i t i o n a l p o r t f o l i o r i s k avers ion (CPRA) at y Q i f his cond i - t i o n a l r i s k y - a s s e t demand at y Q i s p o s i t i v e only i f E(z) > r . His pre - ference i s sa id to d i sp lay g loba l CPRA (GCPRA) i f i t d i sp lays CPRA at a l l y . GCPRA i s stronger than GPRA as GCPRA implies GPRA but the converse i s not t rue . GCPRA i s therefore more r e s t r i c t i v e than GPRA i n the fo l lowing sense: I f for any y , an Investor i s CPRA when p = 1, but not so when p = 0 .5 , then he i s by d e f i n i t i o n GPRA but not GCPRA. N a t u r a l l y we expect to f i n d more people d i s p l a y i n g GPRA than those d i s p l a y i n g GCPRA. According to D e f i n i t i o n 3.9 (3 .8 ) , not a l l investors with p o s i t i v e c o n d i t i o n a l (uncondit ional ) r i s k y - a s s e t demand are c o n d i t i o n a l (uncondi- t i o n a l ) p o r f o l i o r i s k averse, but, i f a GCPRA or GPRA agent invests a p o s i t i v e amount i n the r i s k y a s s e t , i t must be true that E(z) > r . In l i g h t of Theorem U6, i t i s c l ear that any GCCERA investor must also be GCPRA no matter whether h i s preference f u n c t i o n a l i s EU, WU, FDU or LGU. C o r o l l a r y 3.2: Under completeness, t r a n s i t i v i t y and SD, GCCERA implies GCPRA for a c o n d i t i o n a l d i v e r s i f e r . Does GCPRA imply GCCERA i n general or under p a r t i c u l a r preference theories? As i t turns out, GCPRA implies GCCERA for any preference o r - dering s a t i s f y i n g completeness, t r a n s i t i v i t y , s tochast ic dominance, and the c o n d i t i o n a l d i v e r s i f i e r assumption. To show t h i s , we need the fo l lowing lemma: Lemma 3 .1: Suppose x* solves uniquely the CSPC problem (3.4) for a cond i - - 65 - t i o n a l d i v e r s i f e r whose preferences are complete, t r a n s i t i v e , consistent with SD and exh ib i t GCPRA. Then, x* = 0 i f E(z) = r . Proof: Suppose x* < 0. (The d i f i n i t i o n of GCPRA rules out x* > 0.) Then, - x* (> 0) i s optimal for r i s k y asset z' = 2 r - z , c o n t r a d i c t i n g the d e f i - n i t i o n of GCPRA. Q . E . D . Theorem U5.4 (GRA): Under completeness, t r a n s i t i v i t y and SD, GCCERA i s equivalent to GCPRA for a c o n d i t i o n a l d i v e r s i f e r . Proof: Given C o r o l l a r y 2.2, i t su f f i ces to prove GCPRA > GCCERA. Suppose t h e r e e x i s t p e ( 0 , 1 ] , F , H £ Dj such that pFM-( l -p )H >- p6 E ^~^+(l -p)H, where y i s the r . v . a s s o c i a t e d with F . For a given y , construct r = E ( y ) / y and z = i - ( y - y Q r ) + r . Note that E(z) = r . Let x* be the cond i - t i o n a l r i s k y - a s s e t demand i n the CSPC problem with the parameters y , r , z g i v e n above. Then pF ~ +( l -p)H > P F ~ f ( l - p ) H >- po + ( l -p )H. y o y yo Lemma 3.1 impl ies x* = 0, g iv ing r i s e to a c o n t r a d i c t i o n . Q . E . D . - 66 - 4 COMPARATIVE RISK AVERSION In the proceeding sect ions , we character ized r i s k avers ion i n a num- ber of ways. The question to explore next i s : What i s the meaning of one d e c i s i o n maker being more r i s k averse than another? What are i t s behavio- r a l impl i ca t ions? 4.1 D e f i n i t i o n s Since (a) CCE and CE, (b) mps, (c) r i s k y - a s s e t demands, and (d) the concavity of appropriate u t i l i t y functions have been used to character ize GRA, i t i s na tura l to think of them as promising candidates for characte- r i z i n g comparative r i s k avers ion (CRA). We w i l l consider them one by one. (a) C e r t a i n t y Equiva lent Both CCE and CE are concept of a s ing le va lue , therefore can be e a s i - l y extended to a comparative r i s k avers ion context . I f agent A is more r i s k averse than agent B in the sense of CCE (CE) , we w i l l expect agent A to accept a lower CCE (CE) for any d i s t r i b u t i o n than agent B. Formal ly , we say that agent A i s more GCCERA (GCERA) than agent B i f agent A's CCE (CE) of any d i s t r i b u t i o n i s smaller than agent B ' s . (b) Preference Compensated Spread Mean preserving spreads do not work quite wel l i n c h a r a c t e r i z i n g comparative r i s k avers ion . Recognizing t h i s , Diamond and S t i g l i t z (1974) proposed a 'mean u t i l i t y preserving spread' not ion (they c a l l e d i t a 'mean - 67 - u t i l i t y preserving Increase i n r i s k ' ) . S ince , as the name says, th is no- t i o n preserves 'mean u t i l i t y ' , i t can only be used to character ize compa- r a t i v e r i s k aversion for EU-type preferences . For more general p r e f e r - ences, more r e s t r i c t i v e d e f i n i t i o n s are needed. The fo l lowing d e f i n i t i o n i s s i m i l a r to that i n Machina (1982a): D e f i n i t i o n 4 .1: D i s t r i b u t i o n G i s sa id to be a simple compensated spread of d i s t r i b u t i o n F to a dec i s ion maker i f ( i ) G s ing le -crosses F from the l e f t , and ( i i ) F ~ G. Compared with Diamond and S t i g l i t z ' mean u t i l i t y preserving spread ( c f . D e f i n i t i o n 4.5 below), condi t ion ( i ) i n D e f i n i t i o n 4.1 i s more r e s - t r i c t i v e as i t only allows d i s t r i b u t i o n s which cross once. Condi t ion ( i i ) , however, i s more general i n not r e s t r i c t i n g preferences to only expected u t i l i t y ones. Depending on the preferences subscribed to by the d e c i s i o n maker, there are at l east three d i f f e r e n t cases of simple compen- sated spread, i . e . , EU compensated spread, WU compensated spread, and LGU compensated spread. D e f i n i t i o n 4.2: D i s t r i b u t i o n G i s sa id to be a simple mean u t i l i t y preser - v ing spread (simple mups) of d i s t r i b u t i o n F to an expected u t i l i t y d e c i - s ion maker with von Neumann-Morgenstern u t i l i t y funct ion u i f ( i ) G s ing le -crosses F from the l e f t , and ( i i ) EU(F) = EU(G). D e f i n i t i o n 4 .3: For a WU maximizer with value funct ion v and weight func- t i o n w, G i s sa id to be a simple weighted u t i l i t y preserving spread (simple wups) of F i f (a) G s ing le -crosses F from the l e f t ; and - 68 - (b) WU(G) = WU(F). D e f i n i t i o n 4.4: D i s t r i b u t i o n G i s sa id to be a simple LGU preserving spread of d i s t r i b u t i o n F to an LGU dec i s i on maker V i f ( i ) G s ing le -crosses F from the l e f t , and ( i i ) V(F) = V(G) . Since EU i s l i n e a r i n d i s t r i b u t i o n , the squeezed mean i n t e r p r e t a t i o n for second-degree s tochast ic dominance can be general ized to mean u t i l i t y v i a D e f i n i t i o n 4.5 below. This w i l l s i g n i f i c a n t l y increase the set of permiss ib le d i s t r i b u t i o n s . D e f i n i t i o n 4.5: D i s t r i b u t i o n G i s a mean u t i l i t y preserving spread (mups) of d i s t r i b u t i o n F to an EU dec i s ion maker u i f J_ y u(x)dF(x)+u(y)[ l -F(y) ] > J_ y u(x)dG(x)+u(y)[ l-G(x)] for a l l y , (4.1) and u(x)dF(x) = j_2 u(x)dG(x) . (4.2) Condi t ion (4.2) says that d i s t r i b u t i o n s F and G y i e l d equal expected u t i l i t y . Condi t ion (4.1) requires that the 'squeezed' expected u t i l i t y of F over ( - 0 O ,y ] be not less than that of G for a l l y . To consider a sequence of mups of F , Diamond and S t i g l i t z parameter- i zed the d i s t r i b u t i o n with a r i s k fac tor a. D e f i n i t i o n 4.5 can then be res tated as D e f i n i t i o n 4 . 5 ' , where a subscr ipt ind icates the v a r i a b l e with respect to which a p a r t i a l d e r i v a t i v e i s taken. D e f i n i t i o n 4 .5 ' (Diamond and S t i g l i t z ) : Given a d i s t r i b u t i o n F(x ,a ) and a u t i l i t y funct ion u (x ) , an increase i n a represents a mean u t i l i t y p r e - serv ing increase i n r i s k i f T(y) = /JL u ' ( x ) F a ( x , a ) d x = / _ y F a ( x , a ) d u ( x ) > 0 for a l l y , (4.3) - 69 - and T ( » ) = j_l u ' ( x ) F a ( x , a ) d x = j_l F a ( x , a ) d u ( x ) = 0. (4.4) To g a i n some i n s i g h t into D e f i n i t i o n 4 . 5 ' , define F t t = F(x,oc) = (1- a)F+aG, where a e [0,1] and G i s a mups of F as defined by D e f i n i t i o n 4 .5 . F a i s a mixture of F and G with component of G increas ing as a goes from 0 to 1. In o ther words , { F a : a e [ 0 , l ] } represents a sequence of mups of F , going from F towards G. Since the same expected u t i l i t y i s preserved from F to G and F a(x,oc) = ^ [(l-a)F+aG] = G - F , condit ions (4.3) and (4.4) are equivalent to condit ions (4.1) and (4 .2 ) , r e s p e c t i v e l y , noting F ( » ) - G ( » ) = F ( - ° ° ) - G ( - » ) = 0. Suppose that agent A i s more r i s k averse than agent B i n the sense of compensated spread, and that G i s a compensated spread of F to A. It i s expected that G w i l l be preferred to F from B's viewpoint because a less r i s k averse agent should i n general demand less compensation for a given increase i n r i s k . Formal ly , we say that agent A i s more MRA than agent B i f agent A always prefers F to any of agent B's preference compensated spreads of F . (c) Risky-As set Demand In a simple p o r t f o l i o set -up, we defined an agent to be CPRA (PRA) i f h i s c o n d i t i o n a l (uncondi t ional ) r i s k y - a s s e t demand i s s t r i c t l y p o s i t i v e only i f the expected rate of re turn on the r i s k y asset i s s t r i c t l y greater than the r i s k - f r e e rate of r e t u r n . Suppose both agents A and B are CPRA w i t h r e s p e c t to z and r . I f they have i d e n t i c a l i n i t i a l wealth, then i t seems reasonable to expect the more r i s k averse agent to demand less of the r i s k y asset . Formal ly , we say that agent A i s more GCPRA (GPRA) than - 70 - agent B i f f o r any r and z such that E(z) > r agent A's demand for z i s always less than agent B ' s . (d) Concavity of re levant u t i l i t y funct ions Suppose more s tructures are imposed on a preference func t iona l V so that a GCCREA agent must have a concave u t i l i t y funct ion (c f . Theorem LGU5). Agent A being more r i s k averse than agent B suggests that agent A ' s u t i l i t y func t ion , i f i d e n t i f i a b l e , i s 'more concave' than agent B ' s . D e f i n i t i o n 4 .6: An i n c r e a s i n g , continuous funct ion f i s said to be at l ea s t as concave as (more concave than) another i n c r e a s i n g , continuous func t ion g i f there ex i s t s an increas ing concave ( s t r i c t l y concave) funct ion h such that f (x) = h (g (x ) ) . Lemma 4.1 ( P r a t t ) : Suppose f and g are two concave, increas ing funct ions . Then, - f ' 7 f > ( » - g ' V g ' (4.5) i f and only i f f i s at l eas t as concave as (more concave than) g. In words, D e f i n i t i o n 4.6 means that i f f i s more concave than g, then f can be obtained by ' concav i fy ing ' g v i a an increas ing concave funct ion h . For EU max imizers u^ and u^, u^ being more concave than u^, by Lemma 4 .1 , impl ies that - u " / u ! > - u " / u ' . Since - u " / u ' i s the Arrow-Pratt index, A A B B t h i s means that the more r i s k averse an EU i n d i v i d u a l i s , the greater h is Arrow-Pratt index w i l l be. Beyond EU, the 'concavi ty index' w i l l n a t u r a l l y be ~ C " ( x ; F ) / C ( x ; F ) s ince the LOSUF C(x;F) serves as the von Neumann-Morgenstern u t i l i t y - l i k e f u n c t i o n . - 71 - 4.2 Charac ter i za t ions Now that we have c l a r i f i e d the meaning of one agent being more GCCERA, more GMRA, or more GCPRA than another agent, the remaining task i n t h i s sect ion i s to e s t a b l i s h the r e l a t i o n s , i f any, among them. As i t turns out, agent A i s more GCCERA than agent B i f and only i f A i s also more GMRA than B regardless of the preference theory they subscribe to . We s h a l l f i r s t show that th i s i s true for elementary monetary l o t t e r i e s and then extend i t to general monetary l o t t e r i e s . D e f i n i t i o n 4.7: G i s a simple elementary compensated spread of F to a dec i s ion maker i f there ex i s t a, e, p, 9^, 9 2 (e , 9^, 9 2 > 0, p e (0,1]) and an elementary l o t t e r y H such that he i s i n d i f f e r e n t between F = ( l - p ) H + p{i6 a_ e + i 6 a + e } (4.6) and G - ( l - p ) H + p{K-e-9 1 + k + e + 9 2 > ' < 4 ' 7 > In D e f i n i t i o n 4.7, the word 's imple' i s used to i n d i c a t e that G crosses F only once (although the cross ing might be an i n t e r v a l ) . F and G are elementary l o t t e r i e s because they involve a f i n i t e number of out- c o m e s . I n D e f i n i t i o n 4.7, i f 9̂ = 9 2 , G w i l l be an mps of F . For s t r i c t l y r i s k averse dec i s i on makers, an mps of a d i s t r i b u t i o n F i s less d e s i r a b l e than F . In order to make a spread as a t t r a c t i v e , the r i g h t - t a i l s h i f t 9 2 must be greater than the l e f t - t a i l s h i f t 0^. For a s t r i c t l y r i s k seeking i n d i v i d u a l , the opposite i s true ( i . e . 9̂ > 9 2 ) . D e f i n i t i o n 4.8: For e lementary l o t t e r i e s x, £ e J N , £ i s an elementary compensated spread of x i f there e x i s t s a nonnegat ive compensating - 72 - vector rj = ( , TV,,..., t1̂ _-Il ) such that .1 n N-1 _ where x - ( x ^» x2»••*» XN^ zL = ( y 1 » x 2 + T i 1 » x 3 » - - - » x N ) x,n 1 = (?i V i>VVi 'V i>-'V x, = ( y i . - - - . y n - 1 . y n . « t r i . i + \ » x n f 2 * - - x N ) ( 4 - 8 ) Z 1 = ( y 1 , . . . , y n , . . . , y N _ 1 , x N + V l ) = ^ , . . . , y n , . . . . y ^ , y N ) = z . Note that yj 1 ^ and yj 1 are r e s p e c t i v e l y G n _ ^ and given below: Gn-1 = T { E i = l N=2 6 X , } + ¥ { T 6 x +TI + K A 1 * i i n n-1 n+l _ N-2 r n-1 1 . N 1 . i .2,1. 1 , G n " i n E i = l N=2 V + Ei=n+2 N ^ x , ' + N ^ y + 2 6 x ^ + n '* Ji i y n n+l n G n i s a simple elementary compensated spread of Gn_^ i f G^ ~ G ^ _ ^ . D e f i - n i t i o n 4.8 t h e r e f o r e t e l l s us t h a t , i f £ i s an elementary compensated spread of x, then £ can be obtained from x v i a a sequence of simple e l e - mentary compensated spreads i n the fo l lowing manner: S tar t ing with x, i t must be t r u e tha t x^ > y^ . F i r s t push x^ l e f t w a r d s to y^ and push r i g h t w a r d s t o , say , such that the dec i s ion maker's preference i s pre - s e r v e d . L e t us denote the distance of the leftward push at step i by and the d i s t a n c e of the simultaneous rightward push by n^. C l e a r l y , X^= x^-y^> 0, and > x l ~ y i ^ t * i e d e c i s i o n maker i s GMRA. Next, push l e f t w a r d s to , and x^ r i g h t w a r d s to z^ . T h i s t i m e , \^ = z2~y2 = - 73 - ( x 2 + T i 1 ) - y 2 = n 1 + ( x 2 - y 2 ) . In general , \± = \_i+x±~y±> \ > \ > 0 i f t h e d e c i s i o n maker i s GMRA, and y j 1 " 1 = (7 i V i ' W i ' V i ' V z V yj 1 = ( y i » " - . y n - 1 . W r X n ' X n + l + T 1 n ' X n + 2 » - - " X N ) = (yl, • • • . y ^ - L • y n » x n + l + T V i » x n + 2 XN> N - l f o r n = 2, 3, N - l . S i n c e £ must be £ , we have x jj + n jj_j_ = y^» o r V l = y N _ X N > ° ' In l i g h t of the d e f i n i t i o n of GMRA, the fo l lowing c o r o l l a r i e s are obvious: C o r o l l a r y 4 .1: For a r i s k neutra l ( i n the sense of mps) dec i s ion maker, i f £ i s an e l ementary compensated spread of x v i a the compensating vector TJ, then = ^ ^ ( X i - y j ^ ) f o r a l l 1 < k < N - l . C o r o l l a r y 4 . 2 : Suppose £ i s an elementary compensated spread of x for a GMRA d e c i s i o n maker whose p r e f e r e n c e i s comple te , t r a n s i t i v e and c o n s i s t e n t w i th SD. T h e n , the compensat ing v e c t o r TJ s a t i s f i e s the fo l lowing: v > ^ ^ ( X j - y . ^ for a l l 1 < k < N - l . Proof: We prove by i n d u c t i o n . Suppose n^ < x ^ - y ^ . Then SD and GMRA imply that X, 1 = ( y 1 , x 2+n L , x 3 , . . . , x N ) -< ( y 1 , x 2 + ( x 1 - y 1 ) , x 3 x N ) —< (y^+(x^—y^),x 2 ,x 3 , . . . ,x^) = ( x ^ , x 2 , x 3 , . . . , x ^ ) = x, c o n t r a d i c t i n g x ~ y / . Therefore , i t must be true that > x -^-y^. - 74 - k N e x t , s u p p o s e , f o r some k < N , n^ > ( x - £ - y £ ) but \ + i ^ ^ ( x ^ y ^ . Then, k + 1 - ( a. ^ X, - yk'\+l>\+2 fVl' xk+3 X N ) ~ (y1, • • • > v k ' y k + l + X k + l ' x k + 2 ' x k + 3 ' ' * ' » X N ) -< ( y i ' - * - » y k , y k + l + \ + l ' x k + 2 ' x k + 3 ' " * ' x N ^ ^ b y G M R A a n d S D ^ k+1 -< (yi>->->yk>yk+i+h=i(x±-y±)>xk+2>\+3>--'>x^ <by S D ) -< ( y i . - - - » y k » y k + i + \ + ( x k + r y k + i ) » x k + 2 ' x k + 3 V ( b y S D ) = ( y l y k , X k + l + \ , X k + 2 , X k + 3 ' * " , X N ) = Z g i v i n g r i s e to a c o n t r a d i c t i o n . Thus, > ^ L = l ^ x i ~ y i ^ * B y l n ^ u c ~ t i o n , we have proved that > ^jti^i-yi) f o r a l l 1 < k < N. Q . E . D . Lemma 4 . 2 ; Let >^ and >g be the respect ive preference orderings of agents A and B which s a t i s f y completeness, t r a n s i t i v i t y and SD. I f A i s more GCCERA than B, then, for any elementary l o t t e r i e s F and G such that G i s a simple elementary compensated spread of F for A, G >g F . Proof: Since, for A, G is a simple elementary compensated spread of F , by d e f i n i t i o n , there must ex i s t constants a, e, 9̂ , 9 2, p (e, 9̂ , 9 2 > 0, p e (0,1]) and an elementary l o t t e r y H such that F ~^ G and F E ( i - p ) H + p{l 6 a _ £ + l 6 a + £ } , G = ( l - p ) H + p { jVc-9 1 + y 6 a+s+9 2^ Let q^, q 2 e [0,1] be such that q i q i + q ? q? ~ A (i-P) + plA-e-^ + t 1 " "ArX + A + e + e 2 l E Q That B i s less GCCERA than A impl ies G > B Q > B P;">^ F . Q . E . D . Given F defined by (4 .6 ) , Lemma 4.2 implies that , for an i d e n t i c a l downward s h i f t 9^, B w i l l require a lower compensation 0£ than A w i l l . Theorem U7.1 (CRA): For elementary l o t t e r i e s and a pa ir of preference o r d e r i n g s > A and > B s a t i s f y i n g completeness, t r a n s i t i v i t y and SD, the fo l lowing condit ions are equiva lent : (a) For any e l ementary l o t t e r y F , CCE A (F) < C C E B ( F ) , where CCE A (F) and C C E _ ( F ) r e f e r to the CCEs of F ( c o n d i t i o n a l upon any p e [0,1] and an elementary l o t t e r y H) for A and B, r e s p e c t i v e l y . 5 (b) I f £ = G = - ^ ° y i s an elementary compensated spread of x = F E . ^ , 4° to A, then x i s at l eas t as preferable as y to B . i= l N x, ~ v Proof: (a) (b): Since £ i s an elementary compensated spread of x to A, £ can be o b t a i n e d from x v i a a sequence of simple elementary compensated spreads yj 1 given by (4 .8 ) . According to Lemma 4.2 , - O v l v x n v N - l _ (b) -> (a) : This i s s tra ightforward recogniz ing that , for any elementary l o t t e r i e s F and H, pF+(l-p)H i s an elementary compensated spread of ' W 1 - " " ' P.E.D. - 76 - Given Lemma 4.2, Theorem U7.1 i s obvious. Note that , i n condi t ion (b ) , l o t t e r y G need not be a simple compensated spread of F . We are now ready to further extend Theorem U7.1 to general monetary l o t t e r i e s . Theorem U7.2 (CRA): The fo l lowing are equivalent for a pa ir of preference o r d e r i n g s >. and >„ which are complete, t r a n s i t i v e , continuous i n d i s - t r i b u t i o n and consis tent with SD: (a) (GCCERA) For any d i s t r i b u t i o n F , CCE A (F) < C C E B ( F ) , where CCE A (F) and CCE (F) re fer to the CCEs of F for A and B, r e s p e c t i v e l y . B (b) (GMRA) I f G i s a simple compensated spread of an a r b i t r a r y d i s t r i b u - t i o n F to A, then G i s at l east as preferable as F to B. Proof: (a) -»> (b) : Suppose G i s a simple compensated spread of F to A. Consider { i / 2 n : i = 1, 2 n - l }. Let p ° = F(0) and q ° = G(0) . Define i n f {x |F(x)=i /2 n } 0 < i / 2 n < p ° p ° < i / 2 n < p 0 + ( l / 2 n ) p ° + ( l / 2 n ) < i / 2 n < 1, i f p ° i s not i n I = { i / 2 n | i = l 2 n - l } otherwise n sup {X|F(X)=P°} sup {x|F(x)=i/2 n } n _ x „ = and i n f {X|F(X)=P°} i n f t y | G ( y ) = i / 2 n } sup { y J G ( y ) = q ° } sup {y |G(y)=i/2 n } 0 < i / 2 n < q ° p ° < i / 2 n < q ° + ( l / 2 n ) f 0 q°+(l / 2 n ) < i / 2 n < 1, i f q° i s not i n I - { i / 2 n I i = l , . . . , 2 n - l } i n f {y F ( y ) = q ° } otherwise. - 77 - — 9N 1 — 9N—1 C o n s t r u c t F = 2 n 6 n , +TS7 2 ~ n 6 n 1 and G = 2 n 6 n, a +jJL 2~ n6 n, 0 n x. + e i= l x.+e n y « + 9 i= l y . + 6 O n i n J0 n • ' i n such that G ~ G ~ . F ~ F . C l e a r l y , F and G converge i n d i s t r i b u - A n A n A ^ ' n n ° t i o n to F and G, r e s p e c t i v e l y . Furthermore, for n s u f f i c i e n t l y l a r g e , G s ing le crosses F from the l e f t . By Theorem U7.1 , G >D F . n ° n J ' n ~B n To prove G > B F , suppose the contrary that F >- G. Since l i m ^ ^ F ^ = F and l im G = G, by CD, there ex i s t s a K > 0 such that F. >-_ G for n-*» n ' 3 ' k B a l l k > K. Again , by CD, there ex i s t s an M > 0 such that F, >- G for " * » » • ' » k B m a l l m > M. P i c k N = maxlK . M } . Then, F >- G for a l l n > N, g iv ing n B n r i s e to a c o n t r a d i c t i o n . (b) ->• ( a ) : pF+(l-p)H i s a simple compensated spread of p o ^ g (pN +(^-~P)H A to A. Condi t ion (b) impl ies that pF+(l-p)H >g P 6 C C E ( F ) + ( l - p ) H . Since pF+(l-p)H ~ B P 6 C C E ^ ( F ) + ( l - p ) H , by t r a n s i t i v i t y , p 5 C C E * ( F ) + ( 1 - p ) H P 6 C C E ( F ) + ( 1 _ P ) H - SD implies CCE B (F) > C C E A ( F ) . Q . E . D . We have proved that agent A i s more GCCERA than agent B i f and only i f A i s also more GMRA than B regardless of the forms of t h e i r preference func t iona l s as long as they are complete, t r a n s i t i v e , continuous i n d i s - t r i b u t i o n , and consistent with s tochast ic dominance. We next consider comparative GCPRA. It appears that an investor who - 78 - i s more GCCERA than another investor w i l l a lso be more GCPRA no matter what u t i l i t y theory t h e i r preferences subscribe to . Theorem U7.3 (CRA): Suppose the preferences of agents A and B are com- p l e t e , t r a n s i t i v e , continuous i n d i s t r i b u t i o n and consis tent with SD. Then, A i s more GCPRA than B i f A i s more GCCERA than B . P r o o f : In a s imple p o r t f o l i o s e t - u p w i t h E(z) > r , suppose x and x = A B x^+Ax are the respect ive r i s k y - a s s e t demands of A and B who have i d e n t i - c a l i n i t i a l wealth y Q . The o p t i m a l i t y of x^ implies that P F _•_/ . A w ~ N + ( 1 -P ) H >_ pF _ L , ~ . + ( l - p ) H . r y r+(x.+Ax)(z-r) v ~ B v y r+x . ( z - r ) r Jo A Jo A It fol lows that there ex i s t s 9 > 0 such that PF j _ , . A w ~ N +( l -p )H ~ pF ^ , ~ N j » + ( l - p ) H . * y r+(x.+Ax)(z-r) v *' B y r+x,(z-r)+9 v v ' J o A o A Suppose Ax < 0. Then pF , , ~ N l . + ( l - p ) H w i l l be a compensated r r r y r + x . ( z - r ) + 9 v * o A spread of pF . . w ~ s + ( l - p ) H to B . Since A i s more GCCERA than v v y o r + ( x A + A x ) ( z - r ) r B , Theorem U7.2, SD and t r a n s i t i v i t y imply that PF . / . » w ~ N+(1"P)H >. pF , , ~ N . Q + ( 1 - P ) H . v y r+(x.+Ax)(z-r) v ~A v y r+x.(z-r)+9 v v ' Jo A o A >-. pF . , ~ N + ( l - p ) H . A y r f x . ( z - r ) v J o A This contrad ic t s the o p t i m a l i t y of x. to A . Hence, Ax > 0 and x „ > x „ . A B A Q . E . D . Theorems EU7, WU7 and LGU7 below t e l l us that A being more GCPRA than B also impl ies A being more GCCERA than B under EU, WU and LGU. Theorem EU7 (CRA): The fo l lowing are equivalent for a pa ir of continuous, increas ing von Neumann-Morgenstern u t i l i t y functions u^ and u^: (a) (GCCERA) For any d i s t r i b u t i o n F , CCE (F) < CCE ( F ) , where CCE (F) - 79 - and CCE B(F) re f e r to the CCEs of F for u A and u^, r e s p e c t i v e l y . (b) (GCPRA) In a simple p o r t f o l i o set-up with safe asset r and r i s k y a s s e t z, where E(z) > r , suppose u and u* have i d e n t i c a l i n i t i a l wealth and x^ and x^ are t h e i r respective conditional money r i s k y - asset demands. Then x, < x„. A B (c) (Concavity) u i s at least as concave as u . (d) (GCERA) For any d i s t r i b u t i o n F, CE.(F) < CE„(F), where CE.(F) and A B A CE D(F) r e f e r to the CEs of F for u. and u„, r e s p e c t i v e l y . D A. a (e) (GPRA) In a simple p o r t f o l i o set-up with safe asset r and r i s k y a s s e t z, where E(z) > r , suppose u^ and u^ have i d e n t i c a l i n i t i a l wealth and x^ and Xg are t h e i r respective unconditional money r i s k y - asset demands. Then x. < x_. A B Proof: Omitted since t h i s i s well-known. The following lemma i s needed to prove Theorem WU7. Lemma 4.3: I f ( i ) G single-crosses F at x* from the l e f t ; and ( i i ) f(x) and g(x) are two increasing functions, and f(x) i s at least as concave i n x as g(x), P r o o f : Given f ( x ) at l e a s t as concave i n x as g(x), for any x^, x^ £ R such that x n < x„, we have then J[G-F]f'(x)dx > f'(x*) g'(x*) J[G-F]g'(x)dx. f ( x 2 ) g '(x 2) g Applying s t r a i g h t algebra y i e l d s the following: - 80 - £2 [ G - F ] f (x)dx = f* < * * > £ ! [G-F] f^gy dx - * ' (**) U f j G - F ] f g j y dx + j £ [ G - F ] f ^ g j dx} > f»<x*) { f * [ G - F ] dx + / ^ [ G - F ] i ^ O dx} i ; _ c o L g ' ( X * ) V X * L J g'(x*) ' f fx*) 1+00 = f r g r f - J I . [G-F]g ' (x )dx . Q . E . D . I t should be pointed out that the G and F i n Lemma 4.3 need not be d i s t r i b u t i o n funct ions . Nor do f and g have to be re la ted to G and F i n any p a r t i c u l a r way. Theorem WU7 (CRA): Under WU, the fo l lowing are equivalent for two pa ir s of v a l u e and weight functions ( V A » W A ) a n < * ( v B » w g ) with respect ive LOSUF C A and Cg: (a) (GCCERA) F o r any F e D _ , C C E . ( F ) < C C E _ ( F ) , where C C E . ( F ) and J A B A CCE T 1(F) re fer to the CCE of F for ( v . , w . ) and (v_ ,w D ) , r e s p e c t i v e l y . 15 A A a D (b) (GCPRA) In a simple p o r t f o l i o set-up with safe asset r and r i s k y asset z, where E(z) > r , l e t x and x be the respect ive c o n d i t i o n a l A a r i s k y - a s s e t demands of (v ,w.) and (v ,wD) who have i d e n t i c a l i n i -A A D O t i a l w e a l t h . Then x^ < x^ r e g a r d l e s s of the p r o b a b i l i t y and the d i s t r i b u t i o n they are c o n d i t i o n a l upon. (c) ( C o n c a v i t y ) For any F e D j , C^(x;F) i s at l eas t as concave i n x as Cg (x ;F) . In a d d i t i o n , each of the above condit ions implies the fo l lowing: (d) (GCERA) For any F e Dj, C E ^ ( F ) < C E g ( F ) , where CE^(F) and CEg(F) - 81 - re fer to the CEs of F for (v^>w^) a n ^ ( V B ' w g ) » r e s p e c t i v e l y . (e) (GPRA) In a simple p o r t f o l i o set-up with safe asset r and r i s k y asset z, where E(z) > r , l e t x and x be the respect ive r i s k y - a s s e t A B demands of (v ,w . ) and (v ,w ) who have i d e n t i c a l i n i t i a l wealth. A A ii ii Then x. < x „ . A B Proof: (a) -*• (b) follows from Theorem U7.3 . (b) •*• ( c ) : Suppose there ex i s t s H e Dj such that Cg(y;H) i s more concave i n y than C^CyjH). Then, there ex i s t h^ < and q e (0,1) such that C A ( h l ; H ) C l ( h i ; H ) so that , for some 6 > 0, C A (h 2 +9;H)-C A (h 2 ;H) C A ( h 1 ; H)-C A ( h 1 - q 9 ; H ) > 1 ( 4 > 9 ) C B ( h 2 ; H)-C B ( h 2 - 9 ; H ) 1 > ^ ( h ^ m - c ^ h ^ H ) ' ( 4 ' 1 0 ) R e c a l l that J C A ( t ; H ) d [H-F] = ^— WU A[(l-p)F+pH] | . Inequal i ty (4.9) y i e l d s I C A(h i ;H) + \ C A (h 2 ;H) < ~ C ^ - q G j H ) + \ C A (h 2 +9;H) , which i s J C A < t ; H ) d [ H - < ^ + ^ ) ] > / C A ( t ; H ) d [ H - ( ^ h i „ q e + \ \ ^ \ , ^ { w u A [ ( i - P ) ( i 6 h i + \\^-™k\^-*A\-^ \ + B ^ >|p-l > 0. Since ,1, . 1 - 82 - at p = 1, i t i s implied that , for some p s u f f i c i e n t l y c lose to 1, wva-p)4\+ i 5 h 2 ) + p H ] < % t ( 1 - p ) ( k 1 - q e + k 2 + 9 ) + P H ] - < 4 - n > S i m i l a r l y , i n e q u a l i t y (4.10) leads to JB[(i-p)4fihl+ K 2 ) + p H l > ̂ B^^^-^e+T^-e^l - (4.12) WTL L e t y be the common i n i t i a l wealth of WU. and WU,,. Construct a safe J o A B _ ! h 1 +qh 2 ^ _ t L a s s e t r = r-i and a r i s k y asset z = -=-6, , + -=-6, , • Let x. and y Q 1+q 2 h 1 / y Q 2 h ^ A x be the ir respect ive c o n d i t i o n a l demands for z ( c o n d i t i o n a l upon p and B H ) . Also l e t M e(l+q). . M . e(l+q). x l " V1" "t^1 a n d x 2 - V 1 + "V 1̂* Note that x^ < y Q < x,>. It can be v e r i f i e d that (4.11) and (4.12) are equivalent to (4.13) and (4.14) below, r e s p e c t i v e l y : WU A [ ( l -p )F 2+pH] < WU A [ ( l -p )F r + x ( ~ _ r ) + P H ] , (4.13) •'o o 2 WU_[(l-p)F ~tpH] > WU_[(l-p)F . , ~ .+pH]. (4.14) B L y z 1 B y r+x . ( z - r ) ; o o 1 Since both WU. and WU are c o n d i t i o n a l d i v e r s i f e r s , the above amounts to A B x „ < y < x , c o n t r a d i c t i n g condi t ion ( b ) . B o A (c) •*• (a ) : In l i g h t of Theorem U7.3, i t suf f i ces to prove that (c) imp- l i e s that WU. i s more GMRA than WU_. Suppose G i s a simple wups of F to A o WU. and s ing le -crosses F at x* from the l e f t . As C . ( x:F) i s at l eas t as A A concave i n x as C g ( x;F), by Lemma 4 .3 , C ( x * ; F ) J[G-F]C A (x;F)dx > q ( x « | p ) J [G-F]C B (x;F )dx . (4.15) Define F " = ( l -a)F+otG. I t fol lows that 0 = he ^A^ICPO = / y x ; F ) d [ G " F ] " - / [ G-F]C A ( x;F ) d x - 83 - C ! (x* ;F) < A { - | [G-F]C B (x ;F)dx} C ! (x* ;F) n d T T T T . OC. | Cg (x*;F) d £ ''V* ' I O F O ' With CI, CI > 0, th i s implies ^ - W U „ ( F a ) > 0. Hence, WIL(G) > WU„(F) . A D a OL B B B (a) •*• (d) and (b) -> (e) fo l low by d e f i n i t i o n . Q . E . D . Theorem LGU7 (CRA): The fo l lowing are equivalent for a p a i r of LGU func- t i o n a l V A and Vg with LOSUF C A ( x ; F ) and Cg ( x ; F ) , r e s p e c t i v e l y : (a) (GCCERA) F o r any F e D T , CCE . ( F ) < C C E _ , ( F ) , where C C E . ( F ) and J A D A CCE_(F) re fer to the CCEs of F for A and B, r e s p e c t i v e l y . o (b) (Concavity) For any F e D j , C A ( x ; F ) i s at l eas t as concave i n x as Cg ( x ; F ) . I f , i n a d d i t i o n , both V. and are c o n d i t i o n a l d i v e r s i f i e r s , then the A B above condit ions are equivalent to: (c) (GCPRA) In a simple p o r t f o l i o set-up with safe asset r and r i s k y asset z , where E(z) > r , l e t x and x be the respect ive c o n d i t i o n a l A B money r i s k y - a s s e t demand of V . and V„ who have I d e n t i c a l i n i t i a l A B w e a l t h . Then x < x regardless of the p r o b a b i l i t y and the d i s t r i - A a bution they are c o n d i t i o n a l upon. Each of the above condit ions impl ies (e) and (d) below: (d) (GCERA) For any F E D j ( C E A ( F ) < C E g ( F ) , where C E A ( F ) and CEg(F) r e f er to the CEs of F for A and B, r e s p e c t i v e l y . (e) (GPRA) In a simple p o r t f o l i o set-up with safe asset r and r i s k y a s s e t z , where E ( z ) > r , l e t x and x be the respect ive (uncondi- A B t i o n a l ) money r i s k y - a s s e t demand of V. and V_ who have i d e n t i c a l A B - 84 - i n i t i a l wealth. Then x. < x „ . A B Proof: Omitted since i t i s s i m i l a r to the proof of Theorem WU7. A l s o , see Theorem 4 i n Machina (1982a). In th is s ec t ion , we f i r s t defined the meaning of one i n d i v i d u a l being more r i s k averse than another i n d i v i d u a l i n the sense of c e r t a i n t y equiva- l e n t , compensated spread, as wel l as r i s k y - a s s e t demand. We then i n v e s t i - gated i f the comparative r i s k avers ion i n one sense implies the same i n another. Our f ind ing can be summarized as fo l lows . F i r s t of a l l , in order to lay the ground for a meaningful study of comparative r i s k aver- s i o n , i t i s h e l p f u l to impose four propert ies about preference order ings , namely completeness, t r a n s i t i v i t y , cont inu i ty i n d i s t r i b u t i o n and c o n s i s t - ency with s tochast ic dominance. Under these f a i r l y basic assumptions, we were able to demonstrate that "A i s more GCERA than B" <= "A i s more GCCERA than B" <=> "A i s more GMRA than B" => "A i s more GCPRA than B" => "A i s more GPRA than B". This f ind ing i s i n t e r e s t i n g because i t c o n t r a - d i c t s the casual but wide ly-he ld b e l i e f that the equivalence of compara- t i v e GCCERA and GMRA depends on the underly ing preference theory. We are thus convinced that GCCERA and GMRA are more fundamental than prev ious ly thought i n the sense that they are ' theory- independent ' . When we assume l i n e a r Gateaux d i f f e r e n t i a l s to obtain LGU, the Gateaux d e r i v a t i v e C ( x ; F ) , c a l l e d LOSUF (abbrevation for l o t t e r y - s p e c i f i c u t i l i t y f u n c t i o n ) , i s the u t i l i t y funct ion whose degree of concavity serves to measure the degree of r i s k aversion in i t s c o n d i t i o n a l sense. As s t a t e d i n Theorem LGU7, i s more GCCERA than Vg i f f C A ( x ; F ) i s more concave i n x than C „ ( x ; F ) for every F . - 85 - When the func t iona l form of C i s i d e n t i c a l at a l l d i s t r i b u t i o n s , or put d i f f e r e n t l y , when C, i s d i s t r i b u t i o n - f r e e , a l l c o n d i t i o n a l versions of r i s k aversion w i l l reduce to t h e i r uncondi t ional counterparts . The compa- r a t i v e r i s k avers ion for th i s case i s character ized i n Theorem EU7. Another d i s t i n c t i o n between Theorem EU7 and Theorem LGU7 i s the ab- sence of the c o n d i t i o n a l d i v e r s i f e r requirement i n Theorem EU7. This i s because a l l EU maximizers are g e n e r i c a l l y d i v e r s i f e r s . We have known that WU i s a preference theory intermediate between LGU and EU. Comparisions between Theorem LGU7 and Theorem WU7 however reveal few advantages of WU over LGU so far (one being that a l l WU agents are a l so inherent ly d i v e r s i f e r s ) . This does not mean that the a d d i t i o n a l s tructures on WU are i n v a i n . We s h a l l f ind the func t iona l form of WU u s e f u l when we wish to obtain more s p e c i f i c s of CRA. For example, suppose two WU agents happen to have the same value func t ion . I t can be shown that the agent whose weight funct ion decreases fas ter w i l l be more GCCERA. S i m i l a r l y , i f they share the same weight func t ion , the agent with more concave LOSUF w i l l be more r i s k averse . The appeal of the s p e c i f i c func t iona l form i s p a r t i c u l a r l y evident when the problem involves e x p l i c i t opt imizat ion as i n the p o r t f o l i o choice d e c i s i o n (c f . Theorem WU6) or in the study of the normality property of r i s k y - a s s e t demand to be examined i n the next s e c t i o n . - 86 - 5 DECREASING RISK AVERSION AND THE NORMALITY OF RISKY-ASSET DEMAND WITH DETERMINISTIC WEALTH Sect ion 3 i s devoted to studying the behavioral impl i ca t ions of an i n d i v i d u a l ' s r i s k avers ion . In Sect ion 4, we compared two i n d i v i d u a l s and inves t iga ted the impl i ca t ions of comparative r i s k avers ion . Now, l e t us turn back to one s ing le i n d i v i d u a l but allow his i n i t i a l wealth to vary . The questions we attempt to answer are: As an i n d i v i d u a l gets r i c h e r , w i l l he be w i l l i n g to pay a higher or lower insurance premium for a given r i s k ? W i l l h is demand for the r i s k y asset i n our one-safe -asse t -one-r i sky-asse t world increase or decrease? These questions can be answered under d i f f e r e n t assumptions about the r i s k i n e s s of the agent's wealth. In th i s s ec t i on , we continue to assume that the dec i s ion maker has d e t e r m i n i s t i c i n i t i a l wealth. We w i l l al low i t to be s tochast ic i n the next s e c t i o n . This sect ion contains two subsect ions . In Subsection 5.1 , we review the decreasing r i s k avers ion c h a r a c t e r i z a t i o n under EU. Subsection 5.2 focuses on the normali ty of r i s k y - a s s e t demand under WU. 5.1 Decreasing Ri sk Avers ion under Expected U t i l i t y Arrow (1971) convinc ing ly argued for the appeal of decreasing abso- lu te r i s k avers ion which implies that , as an agent gets r i c h e r , he should become less r i s k averse i n the sense of demanding cheaper insurance - 87 - p o l i c i e s and inves t ing more money i n the r i s k y asset . I f the same i n d i v i d u a l with a d i f f e r e n t l e v e l of wealth can be viewed as i f he were a d i f f e r e n t person, then the c h a r a c t e r i z a t i o n of comparative r i s k avers ion (CRA) can be restated s tra ight forwardly to charac ter i ze de- creas ing r i s k avers ion (DRA). EU i s a preference theory under which th i s can be done. Theorem EU8 below i s a d i r e c t t r a n s l a t i o n of Theorem EU7. In accordance with the l i t e r a t u r e , we replace the c e r t a i n t y equivalent condi t ions by the insurance premium ones and state the concavity condi t ion i n terms of Arrow-Pratt index. Theorem EU8 (DRA, Arrow-Prat t ) : The fo l lowing propert ies of a continuous, i n c r e a s i n g , t w i c e - d i f f e r e n t i a b l e von Neumann-Morgenstern u t i l i t y func- t ion u(y) are equiva lent : u"(y Q ) u'Cy^ (a) (Arrow-Pratt Index) - r > - — r - . r- for a l l y < y. • u ' ( y Q ) u ' ( y 1 ) o '1 (b) ( C o n d i t i o n a l Insurance Premium) For any p e (0,1] and H e D , sup- pose n o = i t ( y o , e P , H ) and = n:(y^, e j p ,H) are u 's c o n d i t i o n a l i n s u r a n c e premia for r i s k e at i n i t i a l wea l th l e v e l s y Q and y^, r e s p e c t i v e l y . Then u > for a l l e i f y Q < y^. (c) (Condi t iona l Risky-Asset Demand) In a simple p o r t f o l i o set-up with sa fe a s se t r and r i s k y asse t z , where E(z) > r , suppose Xq and x^ are u 's c o n d i t i o n a l r i s k y - a s s e t demands at i n i t i a l wealth l eve l s y Q and y^, r e s p e c t i v e l y . Then Xq < x^ i f y Q < y^. (d) ( I n s u r a n c e Premium) Suppose Tt = n ( y o , e ) and %̂ = n:(y^,e) are u's i n s u r a n c e premia f o r r i s k e at i n i t i a l wea l th l e v e l s y Q and y^, r e s p e c t i v e l y . Then % > for a l l e i f y Q < y^• - 88 - (e) (Risky-Asset Demand) In a simple p o r t f o l i o set-up with safe asset r and r i s k y asset z , where E(z) > r , suppose Xq and x^ are u's r i s k y - a s s e t demands at i n i t i a l wea l th l e v e l s y Q and y^ , r e s p e c t i v e l y . Then x < x. i f y < y, . o 1 o •'I Theorem EU8 says that , i f an EU dec i s ion maker's preference exh ib i t s decreasing absolute r i s k avers ion , then his c o n d i t i o n a l , as wel l as uncon- d i t i o n a l , insurance premium for any r i s k w i l l decrease and his c o n d i t i o n - a l , as wel l as u n c o n d i t i o n a l , money r i s k y - a s s e t demand w i l l increase as he gets r i c h e r . The asset demand i m p l i c a t i o n is often re ferred to i n the l i t e r a t u r e as the 'normal i ty of r i s k y - a s s e t demand'. Taking Theorem EU8 as a benchmark, we may consider the fo l lowing two g e n e r a l i z a t i o n s . F i r s t , i f we continue to assume that both the i n i t i a l wealth and the wealth increment are de termin i s t i c but adopt more general preference f u n c t i o n a l s , how w i l l the c h a r a c t e r i z a t i o n be modified? I t appears that , beyond EU, the DRA charac ter i za t ions w i l l not be a s t r a i g h t - forward restatement of i t s CRA counterparts . This w i l l be explained s h o r t l y . The second genera l i za t ion of Theorem EU8 is to extend the i n i t i a l wealth or even the wealth increment from de termin i s t i c to s t o c h a s t i c . This w i l l be dealt with i n Sect ion 6. 5.2 Decreasing Ri sk Avers ion and the Normality of Risky-Asset Demand under Non-Expected U t i l i t y Once the u t i l i t y funct ion becomes l o t t e r y - s p e c i f i c , d i f f i c u l t i e s a r i s e i n d i r e c t l y t r a n s l a t i n g CRA charac ter i za t ions to DRA ones. To - 89 - i l l u s t r a t e , consider the uncondi t iona l simple p o r t f o l i o choice problem under WU. In the context of CRA, suppose both investors have the same i n i t i a l wealth y . Let x. and x„ denote the r i s k y - a s s e t demands of C» and Jo A B J A Cg, r e s p e c t i v e l y . By o p t i m a l i t y , we have JC . ( y r + x . ( z - r ) ; F . , ~ . )dF(z ) = 0 (5.1) 1 A o A y r + x , ( z - r ) J o A and J o B In the context of DRA, assume x^ i s C^s r i s k y - a s s e t demand at y^ such that J'C.Cy, r+x, ( z - r ) ; F ^ , ~ N )dF(z ) = 0 . (5.3) J A w l l v ' y^r+x^(z -r ) / v ' v ' I f investor C at y.. can be viewed as investor C at y , we may replace C A i B o A by Cg and y^ by y Q i n (5.3) to obtain (5.4) below: IC , (y r + x . ( z - r ) ; F . , ~ J d F ( z ) = 0. (5.4) J B Jo 1 y r+x, ( z - r ) v v 7 y o 1 The CRA c h a r a c t e r i z a t i o n i n Theorem WU7 can be used to character ize DRA o n l y i f the r e l a t i o n between Xg and y Q i n (5.2) i s i d e n t i c a l to that be- tween x, and y i n ( 5 . 4 ) . T h i s would be the case i f the d i s t r i b u t i o n s 1 •'o t h a t C„ d e p e n d s upon i n ( 5 . 2 ) ( i . e . F , ~ * ) and ( 5 . 4 ) ( i . e . B • y r + x _ ( z - r ) ' v ' v o B ' . ,~ v ) were i d e n t i c a l . Since x„ + x, i n general , we conclude y r + x , ( z - r ) B 1 J o 1 that the DRA c h a r a c t e r i z a t i o n i n terms of asset demand cannot be obtained sim- ply by rephrasing the asset demand condi t ion i n Theorem WU7. This argu- ment a lso appl ies to the insurance premium c o n d i t i o n . Why does the dis tr ibut ion-dependence of LOSUFs cause problems i n c h a r a c t e r i z i n g DRA? In the context of CRA, say i n terms of insurance - 90 - premium, we compare the degree of concavity of two i n d i v i d u a l s ' LOSUFs C^(y;6 _ ) and Cg(y; 6 _ _ ) which are exogenously given and do not s h i f t y o y o B because the d i s t r i b u t i o n s they depend upon remain unchanged. As we turn our i n t e r e s t to DRA, a g a i n , we have two LOSUFs, one depends on F ~ , y o another on F y + ~ . However, these two LOSUFs are not independent of each other because they or ig ina te from the same preference f u n c t i o n a l . The change of an i n d i v i d u a l ' s r i s k a t t i tudes as h is wealth var ie s therefore depends on the movement along £ as wel l as the s h i f t of . This w i l l be explained i n greater d e t a i l a f ter we derive the necessary and s u f f i c i e n t c o n d i t i o n for the normali ty of r i s k y - a s s e t demand under WU. One advantage of WU over LGU i s i t s s t r u c t u r a l s p e c i f i c a t i o n s which enable us to optimize using ca lcu lus and perform comparative s t a t i c s without u t i l i z i n g d i r e c t i o n a l or path d i f f e r e n t i a t i o n . This advantage re su l t s i n Theorems WU8.1 and WU8.2 below: Theorem WU8.1 (DRA and Condi t iona l Risky-Asset Demand): The fo l lowing propert ies of a pa ir of properly s tructured value and weight functions (v,w) with i n c r e a s i n g , concave LOSUF C(x;F) are equivalent: C"(y;G) _ pw'(y) E [C ' (G) ] E [w] (a) p (y ;p ,F ,H) = C'(y;G) (where G = pF+(l-p)H) (5.5) i s decreasing i n y for a l l p e (0,1] and F , H e D^; (b) (Cond i t i ona l Risky-Asset Demand) In a simple p o r t f o l i o set-up with r i s k f r e e asset r and r i s k y asset z = r+n, where E(n) > 0, l e t Xq and x^ be ( v , w ) ' s c o n d i t i o n a l r i s k y - a s s e t demands at i n i t i a l wealth l e v e l s y and y 1 , r e s p e c t i v e l y . Then, x < x. i f y < y , • - 91 - Proof: The CSPC under WU i s Max V(pF r + x ^ - ( l - p ) H ) . x ^o The FOC and SOC for x* to be optimal are as fo l lows: FOC: / C ' ( y * ; G ) T t f F ( T i ) = 0 (3.9) SOC: /C"(y*;G )T ) 2 dF (T i ) < 0, (3.10) where G = pF~ ! f c +(l-p)H and y* = y or+x*r). The rest of the proof i s s i m i l a r to that of Theorem WU8.2. Q . E . D . Although Theorem WU8.1 i s more general than Theorem WU8.2, we e l ec t to present a complete proof of the l a t t e r because i t i s less complicated notat ion-wise and s a c r i f i c e s l i t t l e substance. Theorem WU8.2 (DRA and Risky-Asset Demand): The fo l lowing propert ies of a p a i r of properly s tructured value and weight functions (v,w) with i n c r e a s i n g , concave LOSUF C(x;F) are equivalent: C"(y;F) _ w'(y) (a) p(y;F) * ~ *[C* ™ \ 7 i g V ] (5-6) or e q u i v a l e n t l y , ofv-m - - £ " ( y » F > + w'(y)/E[w] ( ? . P^y,*; - c'(y;F) + C '(y;F)/E[C(F)] i s decreasing i n y for a l l F ; (b) (Risky-Asset Demand) In a simple p o r t f o l i o set-up with safe asset r and r i s k y as se t z = r+n, where E(r)) > 0, l e t X q and x^ be (v,w)'s r i s k y - a s s e t demands at i n i t i a l wealth l eve l s y Q and y^, re spec t ive - l y . Then, x < x. i f y < y, . 1 ' o 1 J o J 1 Proof: The SPC problem under WU i s - 92 - jv(y)w(y)dF ~ ~ Max J r ' — — , where y = y r+xn. Jw(y)dF ' J Jo The FOC and SOC for x* to solve the above are as fo l lows: FOC: / C ' ( y * ; F ) T i d F ( T i ) = 0 (5.8) SOC: / C " ( y * ; F ) T 1 2 d F ( T 1 ) < 0. (5.9) I m p l i c i t l y d i f f e r e n t i a t i n g the FOC (5.8) w . r . t . x* and y Q y i e l d s ,j [ r (F)- . ' ( i rgjg) 1*F _ t ( 5 . i u ; y o - / C ( F ) T i 2 d F where the argument y has been omitted to s impl i fy the express ion. By SOC, the denominator of the RHS of (5.10) i s p o s i t i v e . Therefore , the dx* s ign of —z i s the same as that of d y o J [ C , ' ( F ) - w ' ( / e , / a > d F ) ] T i d F TwdF which can be rewri t ten as E t C " ( y ; F ) - ^ ] - E [ ^ F ) ] E[W ' ( ? ' ) ' ?I ] . ( 5 . i i ) M W J A l s o , r e c a l l that E(z) > r impl ies x* > 0 according to Theorem U6. (a) •*• (b): We need to consider two cases — n > 0 and n < 0. Case ( i ) : n > 0 y = y Q r+x *T i > y Q r . Express ion (5.7) decreasing implies that _ f C"(y;F) _ E [ C ( F ) ] w'(y) > = l C ' ( y ; F ) E[w] C ' ( y ; F ) J p ( y o r ' F ) ~ p o ' M u l t i p l y both sides by - C ' ( y ; F ) T i : C"(y;F)n - w'(y)n E [ ^ ^ ) ] > -C ' (y ;F )np o . (5.12) Case ( i i ) : n < 0 y = y o r+x*n < y Q r . Express ion (5.7) decreasing implies that lC ' ( y ; F ) E[w] C ' ( y ; F ) 1 po' M u l t i p l y i n g both sides by -C ' ( y ; F ) n y i e l d s (5.12) with s t i c t i n e q u a l i t y so that we can take i t s expectat ion as fo l lows: E[C"(y;F)^i] - E[w'(y)r^] E [ ^ f ? ) ] > -p E[C'(y;F)9i] = 0. (by FOC) a [ wj o dx* Hence, -=— > 0. (b) •* (a) : To prove necess i ty , suppose (5.7) i s increas ing at some i n i - t i a l wea l th l e v e l y Q . T h e n , fo l lowing steps i n the s u f f i c i e n c y proof dx* w i l l lead to —— < 0, which contradic t s condi t ion (b) . d y o Q . E . D . The l og i c i n the proof of Theorem WU8.2 i s s i m i l a r to that i n Arrow's (Theorem EU8). We f i r s t t o t a l l y d i f f e r e n t i a t e the FOC for the simple p o r t f o l i o c h o i c e problem to o b t a i n the expression of d x * / d y Q . Then we show that p given by (5.6) decreases i n y i s equivalent to dx*/dy Q > 0. Theorem WU8.2 i s a s p e c i a l case of Theorem WU8.1. Nevertheless , the fo l lowing d i scuss ion w i l l focus on Theorem WU8.2 because i t i s a one-step extension of the now well-known 'normal i ty of r i s k y - a s s e t demand' under EU a t t r i b u t e d to Arrow (1971). By 'one-step' we mean that the only g e n e r a l i - za t ion from Arrow's r e s u l t to Theorem WU8.2 i s the preference f u n c t i o n a l . In contras t , Theorem WU8.1 involves two changes — one i s the preference f u n c t i o n a l , the other i s the i n t r o d u c t i o n of another d i s t r i b u t i o n H. With in the domain of EU, Arrow (1971) showed that , when the i n i t i a l wealth and the wealth increment are both d e t e r m i n i s t i c , an i n v e s t o r ' s pre- ference w i l l d i sp lay decreasing absolute r i s k aversion i f and only i f the s i n g l e r i s k y asse t i s a normal good to him ( i . e . dx*/dy Q > 0 ) . In a d d i - - 94 - t i o n , h is preference w i l l d i sp lay increas ing r e l a t i v e r i s k aversion i f and o n l y i f the safe asset i s a superior (or luxury) good ( i . e . d ( l - B * ) / d y Q > 0 ) . The former r e s u l t i s contained i n the equivalence of (c) and (e) i n Theorem EU8. I t may at f i r s t appear somewhat s u r p r i s i n g that condi t ion (a) i n v"(v) 2w'(v) Theorem WU8.2 i s ne i ther the WU Arrow-Pratt index r (y) = - [ ,y's + — 7 ^ / - ] J v ' ( y ) w(y) J nor the concavity index - C " ( y : F ) / C ( y ; F ) . Given that EU i s a s p e c i a l case of WU, p(y;F) must reduce to the EU Arrow-Pratt index - v " ( y ) / v ' ( y ) when w i s constant. It can be v e r i f i e d that th i s i s indeed so. When w is not constant , ne i ther a decreasing concavity index nor a decreasing Arrow- Prat t index w i l l imply or be implied by a decreasing p(y;F) . We have p a r t l y explained at the s tar t of th i s subsection why a decreasing concav i - ty index i s not the required c o n d i t i o n . As to the Arrow-Pratt index, note that p(y;F) depends on d i s t r i b u t i o n F while the Arrow-Pratt index r (y) does not . This means whether a r i s k y asset i s a normal good to a WU i n - vestor depends on not only h is r i s k a t t i tudes but also the a t t r i b u t e s of the r i s k y a s s e t . To see how the d i s t r i b u t i o n of z a f fec ts one's demand for i t , r e c a l l that C(y;F) = w(y)[v(y)-WU(F)]/ /wdF. (1.25) In the context of simple p o r t f o l i o choice , F i s the d i s t r i b u t i o n of the f i n a l wea l th y = y Q r + x ( z - r ) . C(y;F) i s a weighted u t i l i t y - d e v i a t i o n from WU(F) with w(y)//wdF being the weight. We may i n t e r p r e t C as a weighted regret when v(y) < WU(F) and a weighted r e j o i c i n g when v(y) > WU(F). Let us simply c a l l C (or more s u i t a b l y - £ ) a weighted regret i n genera l . The d e r i v a t i v e of C given below: - 95 - C'(y;F) = w' (y)[v(y)-WU(F)]+w(y)v'(y) /wdF - 7 5 ^ [v(y)-WO(F>] + ^ V ( y ) (1.32) i s the contingent marginal weighted u t i l i t y - d e v i a t i o n from WU(F) or the contingent marginal weighted regret for the outcome y. When y increases by $1, i t causes two e f fects on £ . The f i r s t i s a 'weight e f f e c t ' , given by [w'(y)/E(w)][v(y)-WU(F)] — marginal weight times regre t . The second one i s a ' u t i l i t y e f f e c t ' , g i v e n by [ w ( y ) / E ( w ) ] [ v ( y ) - W U ( F ) ] ' = [w(y)/E(w)]v'(y) — a weighted marginal r e g r e t . The FOC for a WU maximizer's SPC problem i s For an a d d i t i o n a l d o l l a r ' s investment i n z , the extra income i s T) = z - r i f z r e a l i z e s . C'(y*;F)r) i s the marginal u t i l i t y contingent on the r e a l i z a - t i o n of z . The LHS of (5 .8 ' ) gives the expected marginal d i s u t i l i t y from 'bad' outcome states while the RHS gives the expected marginal u t i l i t y from 'good' outcome s ta tes . The FOC means that , at o p t i m a l i t y , the ex- pected marginal u t i l i t y and d i s u t i l i t y from inves t ing an a d d i t i o n a l $1 i n z must balance out so that the agent has no incent ive to deviate from his r i s k y - a s s e t demand x*. Note that i n (5.8) C '(y;F) i s the only term i n v o l v i n g the parameter y . T h e r e f o r e , i n o r d e r to have the r i s k y - a s s e t demand x* r i s e when y ' o o increases , C'(y;F) must behave in a p a r t i c u l a r manner. D i f f e r e n t i a t i n g C'(y;F) w . r . t . y Q y i e l d s the fo l lowing: /C '(y*;F)TidF(n) = 0, (5.8) or e q u i v a l e n t l y , (5 .8 ' ) - 96 - W~ C ( y , F ) " C C y , F ) "JwdF" = r{c"(y;F) - iW- /C'(F)dF} TwdF = r { C ( y ; F ) - E[C'(F)] }. (5.13) The interpretation of p(y;F) defined by (5.6) is better illustrated by expressing it as follows: [-§£- C ' ( y ; F ) ] / r E [ C ' ( F ) ] [_£- C ( y ; F ) ] / V ( y ; F ) P(y;F) = - ° C'(y;F) rE[C*(F)] I {_ £Snll + iM^fl^I} ( 5 6 M ' (F)] 1 C'(y;F) + C'(y;F) >* ; E[C C"(y;F) _ w'(y) _ _ E[C'(F)] E[w] C'(y;F) ( 5 ' 6 ) r E [ C ' ( F ) ] = d W U ( F ) / d y o i s the ex ante expected marginal u t i l i t y from an extra d o l l a r a v a i l a b l e for investment. To be consistent with s tochas t i c dominance, i t must be p o s i t i v e . Since rE[C'(F)] i s constant w . r . t . y, p(y;F) w i l l be decreasing i n y i f and only i f - C' (y;F)] /C'(y;F) (5.14) y o i s also decreasing i n y. Express ion (5.14) has the i n t e r p r e t a t i o n of a y Q - e l a s t i c i t y , i . e . the p r o p o r t i o n a l change i n C ' ( y j F ) induced by $1 increase i n y Q . Note that an increase i n y Q w i l l cause an upward s h i f t of WU(F), changing the benchmark based on which the magnitude of regret i s measured. Thus, the e f fect on C'(y;F) i s twofold, one r e s u l t i n g from the movement along C'» the other from the s h i f t of WU(F). These two ef fects are represented i n (5.13) by the two add i t i ve terms i n the c u r l y bracket . - 97 - p d e c r e a s i n g i n y means that the 'normalized' y Q - e l a s t i c i t y of the m a r g i n a l weighted regret (normalized by oWU(F)/oy o) must be decreasing i n y . A decreasing y Q - e l a s t i c i t y i n turn means that the i n v e s t o r ' s i n t e n s i t y of regret about not obta ining a marginal ly bet ter outcome state lessens as he becomes r i c h e r . This seems reasonable to be the cond i t ion for the nor- m a l i t y of r i s k y - a s s e t demand — as an agent gets r i c h e r , he w i l l hold more of the r i s k y asset . How must the value funct ion v and weight funct ion w behave i n order to d i s p l a y d e c r e a s i n g y Q - e l a s t i c i t y i n C ' (y ;F)? Before we answer th i s quest ion , assume that the SPC problem i s uniquely solved for a r i s k averse dec i s i on maker, i . e . , C ( y ; F ) > 0 and C"(y;F) < 0. The fo l lowing table should be u s e f u l . (a) (b) (c) (d) (e) (f)*** condi t ion for case - C / C w w" w ' / C ( w ' / C ) ' < 0 cond i t ion for p' < 0 (1) 4- 4- (2) 4- 4- (3) 4- + (4) 4- t (5) t 4- (6) t 4- (7) t + (8) + + * s u f f i c i e n t but not necessary for p' < 0 * * * independent of (e) ** necessary but not s u f f i c i e n t for p' < 0 - 98 - - 4- - always + ? - w" < - £.* rill' > J W /E[w] , , w' ^ C l C J l C ( F ) / E [ C ( F ) ] ' ? - ^-r > - 4r-* as above w' C + + impossible as above + - -rill- < -\ W'/E [ w ] r' L C J l C ( F ) / E [ C ( F ) ] J w" C"** + ? - —r < - -5-r as above w C, w" £ " * * - ? - —r > —=r as above w C + + impossible impossible In the above tab le , column (a) spec i f i e s whether £ i s decreas ingly or i n c r e a s i n g l y concave. Since the normali ty of r i s k y - a s s e t demand i s a form of decreasing r i s k avers ion , a decreas ingly concave LOSUF i s the more na- t u r a l case. Columns (b) and (c) ind ica te whether the weight funct ion i s increas ing or decreas ing, concave or convex. When w i s decreasing and concave as i n cases (1) and (5) , the second term of p ( c f . expression (5 .6)) w i l l be decreas ing . On the contrary , i t w i l l be increas ing i f w i s increas ing and convex as i n cases (4) and (8) . For a l l other cases, the d i r e c t i o n i s ambiguous. The condi t ion under which w ' / £ ' w i l l be decrea- sing i s given i n column (e ) . When both -C'7 C' and w'/C' are decreasing i n y, p w i l l d e f i n i t e l y decrease i n y as w e l l . In t h i s category i s case (1) as wel l as cases (2) and (3) when r e s t r i c t e d by the a d d i t i o n a l condi t ion given i n column (e) . In case (8) , the agent's preferences d i sp lay increas ing r i s k avers ion i n the sense of r i s k y - a s s e t demand, i . e . he w i l l reduce his investment in z as he becomes wealthier — a case normatively not very appeal ing . When - C"/C decreases but w'/C' increases , or -C"/C increases but w'/C' de- creases , the decreasing term must dominate i n order to have the normal i ty r e s u l t . I t i s i n t e r e s t i n g to note that even i f the LOSUF C, i s i n c r e a s i n g - l y concave, normal i ty i s s t i l l a t t a i n a b l e . An i n c r e a s i n g l y concave C means that the agent w i l l be more GCCERA or GMRA as he gets r i c h e r . How can we j u s t i f y such an agent's demanding more of the r i s k y asset when he has more money to invest? Since th i s w i l l never occor under EU, l e t us consider an EU agent as our base case. When the weight funct ion i s constant, the second term of p vanishes and p reduces to - 99 - v"(y ) /E[v ' ] v ' ( y ) " S i n c e E [ v ' ] i s constant i n y , p decreasing i n y i s equivalent to - v " / v ' EU d e c r e a s i n g i n y . The l a t t e r i s of course the EU Arrow-Pratt index. i s a normalized concavity index whose behavior happens to be consistent with the Arrow-Pratt index. When w i s not constant, i t i s c l ear from (5.6) that a concave and/or decreasing w w i l l r e i n f o r c e the decreasing r i s k avers ion captured by a de- creas ing normalized concavity index. On the contrary , a convex and/or i n - creas ing w w i l l o f f se t a l l or part of i t . R e c a l l that a decreasing w i n - d icates pessimism which i s a source of r i s k avers ion . A concave, decrea- s ing w therefore depicts decreas ingly pes s imi s t i c a t t i t u d e s . For a WU agent whose u t i l i t y - b a s e d r i s k aversion is increas ing i n wealth ( i . e . [-C'VC]' > 0 ) » If h is pessimism decreases s u f f i c i e n t l y fast when he be- comes r i c h e r , he might s t i l l increase his holding i n the r i s k y asset . I n t e r e s t i n g l y , i f we define a propor t iona l vers ion of p(y;F) as fo l lows: i t can be shown that d ( l - B * ) / d y Q > 0 ( i . e . the safe asset i s a superior good) i f and only i f p*(y;F) increases i n y . Can we obtain a s i m i l a r condi t ion for the normality of r i s k y - a s s e t demand under LGU? The presence of the weight funct ion i n (5.6) apart from C(y;F) leads us to be l ieve that th i s i s not poss ib le without imposing more s tructures on the f u n c t i o n a l V(F) or making further assumptions. I t i s worth noting that the insurance premium condi t ion i s absent i n P*(y;F) = yp(y;F) yC"(y;F) _ yw'(y) E [ C ( F ) ] E[w] C'(y;F) (5.15) - 100 - Theorems WU8.1 and WU8.2. There appears to be some fundamental d i s t i n c - t ions between insurance premium and p o r t f o l i o choice . F i r s t of a l l , note that the absolute s ize of the r i s k i n the insurance premium condi t ion r e - mains the same when the agent's i n i t i a l wealth changes. In contras t , an i n v e s t o r ' s r i s k y - a s s e t demand i s a funct ion of h is i n v e s t i b l e wealth. As he experiences an exogenous increase i n his i n v e s t i b l e funds, he w i l l change his investment i n the r i s k y asset . Suppose his i n i t i a l wealth i n - creases from y Q to yQ+Ay and his r i s k y - a s s e t demand changes from Xq to x^. As long as Xq * x^ , the a b s o l u t e s ize of the r i s k he i s bearing w i l l be a l t e r e d . We also cannot be sure that the r e l a t i v e s ize of the r i s k w i l l not vary . Secondly, as pointed out e a r l i e r , insurance premium i s a consequence of a dec i s ion maker's perception about the c e r t a i n t y equivalent for a r i s k . In contras t , r i s k y - a s s e t demand i s the r e s u l t of an opt imiz ing behavior . As such, the d e r i v a t i o n of the condi t ion for PRA often makes use of the FOC i f i t i s obta inable . We have appl ied the same approach i n producing Theorems WU8.1 and WU8.2. A n a t u r a l way of d e r i v i n g the condi t ion for DRA i n the sense of insurance premium i s the comparative s t a t i c technique. T o t a l l y d i f f e r e n - t i a t i n g WU(F + ~) = WU(6 = v(y o - ix) y i e l d s o o JC'(F. v ' ( y -it) J o (5.16) DRA i n the sense of insurance premium c a l l s for dii/dy < 0, i . e . , / C f ( F ~)dF ' ~ > v ' ( y - n ) . J y +e y +e w o Jo Jo (5.17) - 101 - Under EU, (5.17) reduces to Ju'dF ~ > u'(y -it) J y +e w o o which holds for a l l e i f and only i f -u"/u' i s decreasing. When the pre- ference functional i s nonlinear i n d i s t r i b u t i o n , (5.17) w i l l not be equi- valent to (-C/C')' < 0 i n general. - 102 - 6 COMPARATIVE AND DECREASING RISK AVERSION INVOLVING STOCHASTIC WEALTH In Sect ion 5, both the i n i t i a l wealth and the wealth increment are assumed to be d e t e r m i n i s t i c . When complete insurance i s not a v a i l a b l e or when a safe asset does not e x i s t , th i s assumption i s deemed u n r e a l i s t i c . I t i s therefore of in t ere s t to see how the CRA and DRA charac ter i za t ions i n Theorem EU7 and Theorem EU8 can be extended to al low for s tochast ic i n i t i a l wealth, or even s tochast ic wealth increments. It should be pointed out that the wealth increment may confound the a g e n t ' s p o r t f o l i o choice problem. To i l l u s t r a t e , l e t r and z = r+n with E ( n ) > 0 be the two a s s e t s i n our s imple p o r t f o l i o set -up. Suppose an agent's demand for z i s Xq when his r i s k y i n i t i a l wealth i s y . His f i n a l wealth w i l l be y r+x n. I f his demand for z changes to x, = x +Ax (Ax may Jo o 1 o J be p o s i t i v e or negat ive ) when his i n i t i a l wealth increases to y^ = yQ+Ay (Ay > 0 ) , h is f i n a l wealth w i l l be (y o+Ay)r+(x o+Ax) T) i f Ay i s i n v e s t i b l e , and yQr+(x^+Ax)n+Ay i f Ay i s not i n v e s t i b l e . When Ay i s i n v e s t i b l e , Ax i s a consequence of two e f f e c t s . One may be c a l l e d a 'resource e f f e c t ' — an ef fect caused by an increase in his i n v e s t i b l e resources . The other may be c a l l e d a ' r i s k a t t i tude e f f e c t ' — an e f fect caused by the change i n his a t t i tudes toward r i s k which i n turn i s caused by an increase i n his 'consumable' income. We s h a l l only be concerned with the r i s k a t t i tude e f fect i n th i s s ec t ion . In other words, we assume that the ant i c ipa ted wealth increment w i l l be a v a i l a b l e only - 103 - af t er the investment dec i s ion i s made, therefore non inves t ib l e . For sim- p l i c i t y , we say the wealth increment i s ex post . In contras t , the unres- t r i c t e d wealth increment i n Theorems EU8, WU8.1, and WU8.2 i s ex ante. Note that , when wealth increment i s ex ante but can only be invested i n the safe asset , the r e s u l t i n th i s sec t ion s t i l l ho lds . This sec t ion contains two par t s . In Subsection 6 .1 , Theorem EU9 and Theorem EU10, due to Ross (1981), extend Theorem EU7 and Theorem EU8, r e s - p e c t i v e l y , to al low for s tochast ic i n i t i a l wealth. In Theorem EU10, which i s i n terms of DRA, the wealth increment i s assumed d e t e r m i n i s t i c . In Subsection 6.2, we f i r s t i l l u s t r a t e that Ross' strong concavity index does not have a WU or LGU counterpart . Theorems LGU9 and LGU10 are then presented as s p e c i a l cases of Machina (1982b)'s Theorem 1 extended to LGU by imposing a d d i t i o n a l s tructure on the l i n e a r Gateaux d e r i v a t i v e C (Chew, 1985). Aga in , we assume s tochast ic i n i t i a l wealth and determinis - t i c wealth increments. Both i n i t i a l wealth and wealth increment are allowed to be s tochas t i c in Theorem LGU11. This case apparently involves too many r i s k s for EU to handle. Hence, there i s no Theorem EU11. Since the r e s u l t s gathered i n th i s sect ion are e i ther from Ross (1981) or based on Machina (1982b) and Chew (1985), t h e i r proofs w i l l be discussed but not reproduced. The presence of th i s sect ion i s mainly for the completeness of comparisons among EU, WU and LGU under d i f f e r e n t assumptions about wealth l e v e l s , namely from ( y Q , A y ) , to ( y Q , A y ) , and then to ( y Q , A y ) . - 104 - 6.1 CRA and DRA with Stochast ic Wealth under Expected U t i l i t y Suppose two EU agents u^ and Ug have the same r i s k y i n i t i a l wealth y . L e t H a and be t h e i r respect ive insurance premia for a r i s k e un- c o r r e c t e d with y . Let x A and Xg be t h e i r respect ive r i s k y - a s s e t demands i n a s imple p o r t f o l i o s e t - u p with assets r and z = r + r i , where E(r f | y Q ) = E(rf) > 0 f o r a l l y . I f u^ i s more r i s k averse than Ug, we expect u^ to be w i l l i n g to pay a h i g h e r premium than u for the insurance against a B g i v e n r i s k e. S i m i l a r l y , we a n t i c i p a t e Ug to invest more i n the r i s k y a s s e t than u w i l l . What i s the proper condi t ion for being 'more r i s k A. averse' i n th i s sense? The answer i s given i n Theorem EU9 below: Theorem EU9 (CRA wi th y , R o s s ) : The f o l l o w i n g propert ies of a pa ir of continuous, s t r i c t l y i n c r e a s i n g , t w i c e - d i f f e r e n t i a b l e von Neumann- Morgenstern u t i l i t y funct ions u A and Ug are equivalent: u A(y+k) Ug(y+k) (a) (Strong Arrow-Pratt Index) - — , , N > - — , , . (6.1) u A ( y ) u B ( y ) for a l l k. (a ' ) (Strong Concavity) There ex i s t a p o s i t i v e constant \ and a decrea- s ing concave funct ion h such that u A (y ) = ^Ug(y) + h ( y ) . (6.2) (b) ( I n s u r a n c e Premium) Suppose it and % are the respect ive insurance A B premia for r i s k e of u. and u„ who have i d e n t i c a l i n i t i a l wealth y . r A B o Then, \ > \ f o r a 1 1 ^ s a t i s f y i n g E ( e | y Q ) = E(e) for a l l y Q . In a d d i t i o n , each of the above implies the fo l lowing cond i t i on : (c) ( R i s k y - A s s e t Demand) Suppose u A and u f i have i d e n t i c a l wealth y Q and - 105 - x A , X g are t h e i r respect ive r i s k y - a s s e t demands i n a simple p o r t f o - l i o set-up with assets r and z = r + T ) , where E(rj jy Q ) = E(t)) > 0 for a l l y . Then, x. < x „ . 3 o ' A B Note that the r i s k y - a s s e t demand condi t ion i s implied but not equiva- lent to the other c o n d i t i o n s . This i s another evidence that the nature of insurance premium and r i s k y - a s s e t demand i s not quite the same. In th i s case, x A < X g impl ies 0 > EIuj^r+XgTOri] = E [ X u g ( y o r + X g T i ) T i ] + E[h ' (y^r+j^ T ) ) TI] = E[h ' (y r+xJn)TJ] O D - C o v [ h » ( y r+x ^),^] + E[h» (y r+xJn)]E[^n] O D O a which may be s a t i s f i e d by a funct ion h not simultaneously decreasing and concave. The strong Arrow-Pratt index condi t ion in Theorem EU9 i m p l i e s , but i s not implied by, the Arrow-Pratt index condi t ion i n Theorem EU8. Due to i t s stronger form here, the i n i t i a l wealth i s allowed to be random. The randomness i s however not a r b i t r a r y . It must be uncorre lated to the r i s k e to be insured or the r i s k n from the r i s k y asset . To gain some ins igh t into Ross' strong Arrow-Pratt index, consider an i n f i n i t e s i m a l r i s k e which i s contingent on the r e a l i z a t i o n of the i n i t i a l wealth y . Suppose e i s l i k e l y to occur only i f y Q e Y and w i l l d e f i n i t e - l y not occur i f y Q e Y; E ( e | y o ) = 0 and Var(e ' jy o ) = a 2 for every y Q e Y. M o r e o v e r , l e t F and G be the d i s t r i b u t i o n s of y Q and y Q + £ > r e s p e c t i v e l y . The insurance premium it for th i s contingent r i s k i s defined by equation (6.3) below: - 106 - Ju(y o-n)dF = J Yu(y O + E)dG + /_u(y Q)dF. (6.3) Expand both sides v i a Taylor's series as follows: Ju(y o - T t)dF = J[u(y o)-mi'(y o)+0(Tt 2)]dF = J u ( y Q ) d F - n;Ju'(y o)dF + 0 ( n 2 ) . / Yu(y o+£)dG + J_u(y o)dF = J Y[u(y o)+eu'(y o)+ ^ u"(y Q)+o(e 2)]dG + J_u(y Q)dF = J y u ( y o ) d F + ^ J Y u " ( y o ) d F + /_u(y Q)dF + o( a 2) = J u ( y Q ) d F + ^ / Yu"(y Q)dF + o ( a 2 ) . Therefore, a 2 V ' < y o ) d F * y o » e ) " - T - J u ' ( y o ) d F ' ( 6 - 4 ) The term / Y u " ( y o ) d F i s an expected diminishing rate of the marginal u t i l i t y which measures the d i s u t i l i t y from $1 uninsured l o s s . Since the r i s k £ w i l l l i k e l y be present only at y Q £ Y, the i n t e g r a l i s taken over the set Y only. The term -/u'(y Q)dF on the other hand gives the expected d i s u t i l i t y from paying one extra d o l l a r i n premium for the insurance. The e x p e c t a t i o n i s taken over the union of Y and Y because the premium has to be paid no matter y £ Y or y e Y. The r a t i o of these two terms i s the r J o Jo modified Arrow-Pratt index for the case where the agent's wealth i s sto- c h a s t i c , and has the i n t e r p r e t a t i o n of "twice the insurance premium per unit of conditional variance" of the i n f i n i t e s i m a l contingent r i s k to be insured. For two EU maximizers u. and u_, with i d e n t i c a l y d i s t r i b u t e d as F, A B Jo * we have %^ > %^ for a l l such r i s k s only i f - 107 - W V d F > « V / u A ( y Q ) d F / u B ( y Q ) d F { b - 5 > f o r a l l Y . Furthermore, under EU the r i s k e can be general ized to nonin- f i n i t e s i m a l ones. It i s s tra ightforward to check that (6.5) holds for a l l F ~ and Y i f and only i f y o u A(y+k) u B(y+k) ~ u A (y ) >_-sp?r (6,1) for a l l k. Condi t ion (a' ) of Theorem EU9 says u A can be obtained by transforming u v i a ( 6 . 2 ) . R e c a l l t h a t , i f u i s more concave than u , then u can be U A 15 A o b t a i n e d by " c o n c a v i f y i n g " u^ v i a an i n c r e a s i n g , concave f u n c t i o n . Does (6.2) require u be even more concave? Given an i n c r e a s i n g , concave func- A t i o n u „ , suppose D u A ( y ) = Au B (y) + h ( y ) , and u A (y ) » h [ u B ( y ) ] , where X > 0 , h' < - X u l < 0 , h" < 0 , h* > 0 and h" < 0 . Then u ! , u! > 0 , B A A u A , u A < 0 and - U A / U A > - U A ^ U A * * n o t ^ e r words, i n order to have the CRA c h a r a c t e r i z a t i o n i n Theorem EU7 carry through to the case with s tochast ic i n i t i a l wealth, the u t i l i t y funct ion of the more r i s k averse i n d i v i d u a l A must be more concave than i n the case where wealth i s d e t e r m i n i s t i c . Note t h a t , because u A and Ug are not l o t t e r y - s p e c i f i c , ne i ther w i l l be X and h ( y ) . This i s c r u c i a l for the proof of Theorem EU9. In the context of DRA, suppose an agent's insurance premium for r i s k e a t i n i t i a l wea l th y i s n . When y i s increased by a constant Ay i n - 1 0 8 - every s ta te , how w i l l h is insurance premium change accordingly? I f he i s decreas ingly r i s k averse, we expect him to become more re luc tant to pur- chase insurance. Thus, the premium he w i l l be prepared to pay for the same p o l i c y should decrease. S i m i l a r l y , we expect him to increase his ho ld ing of the r i s k y asset when he gets r i c h e r . The condi t ion for th i s sense of DRA is given in Theorem EU10 which i s simply a rephrasing of Theorem EU9. Theorem EU10 (DRA with y Q and A y , R o s s ) : The fo l lowing propert ies of a continuous, s t r i c t l y i n c r e a s i n g , t w i c e - d i f f e r e n t i a b l e von Neumann- Morgenstern u t i l i t y funct ion u(y) are equivalent: (a) (Strong Arrow-Pratt Index) - U u t ^ y ^ i s decreasing i n y for a l l k. (b) ( I n s u r a n c e Premium) Suppose I C q = i t (y o > e ) and iz^ = it(y^,e) are u's i n s u r a n c e premia f o r r i s k e at s tochas t ic i n i t i a l wealth l e v e l s y Q and y^ = yQ+Ay, r e s p e c t i v e l y . Then, Ay > 0 impl ies TI > %^ for a l l e s a t i s f y i n g E ( e j y Q ) = E(e) for a l l y Q . In a d d i t i o n , each of the above impl ies the fo l lowing property: (c) ( R i s k y - A s s e t Demand) Suppose Xq and x^ are u's r i s k y - a s s e t demands at i n i t i a l wealth l eve l s y Q and y^ = y Q+Ay, r e s p e c t i v e l y , i n a sim- p l e p o r t f o l i o s e t - u p wi th sa fe a s se t r and r i s k y asset z = r+T), where E ( t i | y o ) = E(r\) > 0 for a l l y Q . Then, XQ < ^ i f Ay > 0. - 109 - 6.2 CRA and DRA with Stochastic Wealth Beyond Expected U t i l i t y To derive the condi t ion for comparative and decreasing GIPRA under WU, we apply the same approach as i n the preceding subsect ion. Consider an i n f i n i t e s i m a l r i s k £ c o n t i n g e n t on y Q e Y w i th E ( e | y Q ) = 0 and V a r ( e j y o ) = a2 f o r every y Q e Y . Let F and G be the d i s t r i b u t i o n s of y Q and y Q +e , r e s p e c t i v e l y . A WU a g e n t ' s i n s u r a n c e premium i s defined by (6.6) below: Jv( y o - n) w ( y o - it) dF / Y v ( y 0 + £ ) w ( y 0 + £ ) d G + J y v ( y 0 ) w ( y Q ) d F Jw(y o-it)dF = J Y w(y O +E )dG+J;£w(y o )dF ( 6 ' 6 ) Expanding both sides v i a Tayor's ser ies a f ter cross m u l t i p l i c a t i o n y i e l d s a2 / Y C ( y 0 ; F ) d F * y o » e ) " ~2 -/C'(y ;F)dF ' ( 6 ' 7 ) In view of the s i m i l a r i t y between (6.7) and (6.4), one i s tempted to speculate that -C"(y+k>F)/C'(y;F) decreasing i n y for a l l k and F i s the c o n d i t i o n we are seeking f o r . This conjecture turns out to be i n c o r r e c t . It i s true that CA(y+k;F) q(y+k;F) CA(y;F) CB(y;F) f o r a l l k and F i f and only i f , for every F E D , there ex i s t a constant > 0 and a decreas ing , concave funct ion h^(y) such that CA(y;F) = ^C B (y ;F) + h p ( y ) . S i n c e C and C are F - s p e c i f i c , X and h must a l s o depend on F . As a A B r e s u l t , the proof of Theorem EU9 w i l l not go through for WU. From (6.7) we know that , i f %̂ i s to be greater than TCg for a l l Y, i t must be true that - 110 - C A(y;F) q ( y ; F ) " /CA(F)dF > " /CB(F)dF ( 6 , 8 ) f o r a l l y , where F i s the d i s t r i b u t i o n of y . Although (6.8) i s derived for small r i s k s , i t turns out s i m i l a r to Machina's condi t ion for general r i s k s . To extend Machina's r e s u l t s from FDU to LGU, we impose a d d i t i o n a l s tructure on LOSUF C as below (Chew, 1985): Assumpt ion 6 . 1 : The l i n e a r Gateaux d e r i v a t i v e £ ( • ; • ) : J * Dj ^ R of the preference func t iona l V ( » ) : D^ ^ R i s continuously d i f f e r e n t i a b l e and ex ante bounded, i . e . , there ex i s t s M > 0 such that | c ( x ; F ) | < M for a l l x e J and F e D j . In th i s s ec t ion , we suppose Assumption 6.1 i s s a t i s f i e d . In what fo l lows , the theorems are stated i n terms of LGU only . Theorem LGU9 (CRA with y ): The fo l lowing propert ies of two LGU f u n c t i o n - a l s V . , V with i n c r e a s i n g , concave LOSUFs C and C s a t i s f y i n g Assump- A 15 A i5 t l o n 6.1 are equiva lent : C A (y;F) q ( y ; F ) ( a ) " /CA(F)dF > " /CB(F)dF ( 6 ' 8 ) for a l l y and F . (b) ( I n s u r a n c e Premium) Suppose u A and %̂ are the respect ive insurance premia for r i s k £ of C. and C_ who have i d e n t i c a l i n i t i a l wealth y . A B •'o Then, \ > \ f o r a 1 1 ^ s a t i s f y i n g E ( e j y o ) = E ( £ ) at a l l y . In a d d i t i o n , i f both V. and V_ are d i v e r s i f e r s , then each of the above A B i s equivalent to: (c) ( R i s k y - A s s e t Demand) Suppose C. and C„ have i d e n t i c a l wealth y and A B O - I l l - , Xg are t h e i r respect ive r i s k y - a s s e t demands i n a simple p o r t f o - l i o s e t - u p with safe asset r and r i s k y asset z = r+n, where E ( n | y o ) = E(TI) > 0 for a l l y . Then, x. < x,,. O A B Theorem LGU10 (DRA wi th y Q and A y ) : The fo l lowing propert ies of an LGU f u n c t i o n a l V with i n c r e a s i n g , concave LOSUF C s a t i s f y i n g Assumption 6.1 are equivalent: ( a ) - g"(y;F) > _ C"(y*;F*) r , . ; /C'(F)dF /C '(F*)dF* for a l l y* > y and F*(s) = F ( s-A), A > 0. (b) ( I n s u r a n c e Premium) Suppose % and are C's insurance premia for r i s k e at i n i t i a l wea l th l e v e l s y Q and y^ = y Q+Ay, r e s p e c t i v e l y . Then, \ > \ f o r a l l ^ s a t i s f y i n g E ( e y Q ) = E ( e ) for a l l y Q i f Ay > 0. In a d d i t i o n , i f V is a d i v e r s i f i e r , then each of the above i s equivalent to the fo l lowing: (c) ( R i s k y - A s s e t Demand) Let Xq and x^ be C's respect ive r i s k y - a s s e t demands at i n i t i a l wealth l eve l s y^ and y^ = yQ+Ay i n a simple p o r t - f o l i o s e t - u p wi th safe a s se t r and r i s k y as se t z = r+rj, where E ( r i | y o ) = E(TJ) > 0 for a l l y Q . Then, XQ < *1 i f Ay > 0. Theorem LGU11 (DRA wi th y Q and A y ) : The fol lowing propert ies of an LGU f u n c t i o n a l V with i n c r e a s i n g , concave LOSUF C s a t i s f y i n g Assumption 6.1 are equiva lent : / n x ... C"(y;F) _ C ( y * ; F * ) . ( a ) /C (F )dF > /C*(F*)dF* ( 6 ' 1 0 ) for a l l y* > y and F* > i F . - 112 - (b) ( I n s u r a n c e Premium) Suppose TI and it̂ are C ' s insurance premia for r i s k e at i n i t i a l wealth l eve l s y Q and y^ = y^+Ay, r e s p e c t i v e l y , and Ay > 0. T h e n , % Q > \ f o r a 1 1 ^ s a t i s f y i n g E ( e j y o ) = E(ejy o +Ay) = E(e) for a l l y and y +Ay. Jo Jo In a d d i t i o n , i f V i s a d i v e r s i f e r , then each of the above condit ions i s equivalent to: (c) ( R i s k y - A s s e t Demand) Suppose X q and x^ are C ' s respect ive r i s k y - a s s e t demands at i n i t i a l wealth l eve l s y and y, = y +Ay > y i n a s imple p o r t f o l i o se t -up with safe asset r and r i s k y asset z = r+T), where E ( T ) j y o ) = E(ri |y o +Ay) = E(TI) > 0 for a l l y Q and yQ+Ay. Then X Q < x r Several points are worth noting i n the above theorems. F i r s t , the r i s k £ i n i n s u r a n c e premium c o n d i t i o n and the r i s k n i n the r i s k y - a s s e t demand c o n d i t i o n are required to be uncorre lated with y Q and Ay i n such a manner that E ( e | y o ) = E ( £ | y o + A y ) = E ( E ) and E(r ) | y o ) = E(Tl | y o + A y ) = E(TJ) > 0 f o r a l l y Q and y Q +Ay. When his wealth i s r i s k y , an agent's insurance premium and r i s k y - a s s e t demand w i l l n a t u r a l l y depend on how d i f f e r e n t sources of r i s k s i n t e r a c t . The purpose of the uncorrelatedness r e s t r i c - t i o n i s to e l iminate any poss ib le o f f s e t t i n g or aggravating e f fec t among r i s k s . Secondly, the r e s u l t i n Theorem LGU11 was o r i g i n a l l y proved by Machina for Fre"chet d i f f e r e n t i a b l e u t i l i t y . E s s e n t i a l l y , the proof i s c o m p a r a t i v e s t a t i c s u t i l i z i n g path d e r i v a t i v e . For example, i f F ~ ~ i s y o i n d i f f e r e n t to F ~ , then there ex i s t s a path from F ~ to F ~ ~ along y - i r y - i t y +E J o o o o Jo - 113 - which the same u t i l i t y l e v e l i s maintained. In other words, th i s path i s an ind i f f erence curve. Consider a scenario s i m i l a r to Kahneman and Tversky (1979)'s p r o b a b i l i s t i c insurance. Suppose an agent can purchase an i n s u r a n c e for e at a premium it( a) lower than the complete insurance premium If the r i s k e occurs , l o t s w i l l be drawn to determine whether the insurer or the insured i s to absorb the r i s k i n the event i t occurs . With a chance, the insurer w i l l be respons ible for e; with 1-a chance, the in surer w i l l walk away with the premium, leaving the agent to absorb the r i s k . N a t u r a l l y , n:( a) increases i n a. A l o n g the ind i f f erence path {F a=aF~ , N + ( l - a ) F ~ , N , -} , ^ - V ( F a ) & y -it(a) y -it(a)+e J ' da v ' J o o = 0 at a l l a e [ 0 , 1 ] . S i m i l a r l y , g i v e n Ay > 0, t h e r e i s another i n d i f f e r e n c e path {F* a=aF~ ~ N + ( l - a ) F ~ , ~ N~} . If the agent v L y o +Ay-n*(a) v ' y o+Ay-it*( a)+eJ & i s decreas ingly r i s k averse, then %(a) > n*(a) at each a e [0 ,1 ] . Define a path JF a =aF~ , ~ N + ( l - a ) F ~ . . ~ , N . ~} . Since it( a) i s too high to be r 1 y Q+Ay- n;(a) yQ+Ay-n;( a)+e' v ' & o p t i m a l f o r wea l th yQ+Ay, a lower a w i l l be p r e f e r r e d . Hence ^ V ( F a ) < 0. When C s a t i s f i e s Assumption 6 .1 , V i s path d i f f e r e n t i a b l e on a l l general ized smooth paths (see Chew (1985)). As such, Machina's re su l t s w i l l continue to h o l d . R e c a l l that the CRA condi t ion we derived for i n f i n i t e s i m a l r i s k s i s : q (y;F) C B ( y ; F ) > ~ trx /t.\ji7 (6.8) / C A ( F ) d F / C B ( F ) d F for a l l y and F . In the context of DRA, we must take into cons iderat ion the e f fec t on the r a t i o caused by the s h i f t of d i s t r i b u t i o n . Af ter a - 114 - d e t e r m i n i s t i c increase i n ex post wealth Ay, the d i s t r i b u t i o n of the f i n a l wealth w i l l be F ~ . The condi t ion for DRA therefore becomes y+Ay. / C ( F ~ ">dF~ (6-11) J g y+Ay ; y+Ay decreasing i n both y and Ay. This i s the case of Theorem LGU10. When the wealth increment is s tochas t i c , i t i s required that the r a t i o (6.11) decrease i n y as wel l as i n d i s t r i b u t i o n i n the sense of f i r s t - d e g r e e s tochas t ic dominance. Hence, condi t ion (a) of Theorem LGU11. Obviously (6.10) impl ies (6 .9 ) . To see the d i s t i n c t i o n between them, c o n s i d e r a p o s i t i v e s t o c h a s t i c wea l th i n c r e a s e Ay = A+9 w i t h A > 0, E( 0 y ) = 0 f o r a l l y and 6 i s bounded from below by - A . Consider an o o agent who i s decreas ingly r i s k averse i n the sense of Arrow. As h i s wealth increases by a p o s i t i v e , de termin i s t i c amount A, he w i l l become less r i s k averse — the i m p l i c a t i o n of condi t ion (6 .9 ) . When an uncorre- c t e d , zero-mean r i s k i s added to h is wealth, he w i l l f e e l worse-off , therefore become more r i s k averse. The net e f fect of an uncorre lated zero-mean r i s k and a simultaneous de termin i s t i c increase i n wealth however i s ambiguous i n genera l . Condi t ion (6.10) i s stronger than (6.9) i n the sense that i t further requires that the ef fect on an agent's r i s k a t t i t u d e caused by A not be o f f se t by the opposite e f fect of any zero-mean r i s k 9 bounded from below by - A . This stronger measure can be rephrased to character ize CRA for the case where agents have i n d e n t i c a l s tochast ic i n i t i a l wealth y . In such a case, agent V. i s sa id to be more r i s k averse than agent V,, up to A i f C A (y;F) Cg(y;G) /C A (F)dF /C B (G)dG - 115 - f o r a l l y , 9 and F , where 9 s a t i s f i e s E(9) = 0 and min{9} > -A and G i s the d i s t r i b u t i o n of s+9 i f F i s that of s. Also note that there i s no Theorem EU11. This i s because the von Neumann-Morgenstern u t i l i t y funct ion u(y) does not depend on d i s t r i b u t i o n , rendering EU incapable of handling the s i t u a t i o n where both i n i t i a l wealth and wealth increment are s t o c h a s t i c . On the other hand, Theorems WU9, WU10 and WU11 are omitted because they w i l l be i d e n t i c a l to the i r LGU counterparts without the d i v e r s i f i e r requirement. - 116 - 7 CONCLUSION A f t e r an extended period of the predominance of expected u t i l i t y i n economics and f inance , there i s a sense of excitement i n terms of new d i - rec t ions being contemplated. D e s c r i p t i v e v a l i d i t y has provided the p r i - mary impetus behind many attempts to construct theories beyond expected u t i l i t y . They include the theories of A l l a i s (1953; 1979), Edwards (1954), Handa (1977), Meginniss (1977) and Karmarkar (1978), Kahneman and Tversky's prospect theory (1979), the regret theory of B e l l (1982) and Looms and Sugden (1983), Machina's Fre*chet d i f f e r e n t i a b l e preference func- t i o n a l analys i s (1982a; 1982b) and weighted u t i l i t y (Chew and MacCrimmon, 1979a; 1979b; Chew, 1980; 1981; 1982; 1983; F i shburn , 1983; Nakamura, 1984). In order to d i scr iminate among these a l t e r n a t i v e preference theo- r i e s , further experimental studies w i l l be needed to de l ineate t h e i r r e s - pect ive domains of empir i ca l v a l i d i t y . Another way of d i s c r i m i n a t i n g among them i s v i a t h e i r a p p l i c a b i l i t y to the economics of uncerta inty and informat ion . In comparison with ex- pected u t i l i t y , few such a p p l i c a t i o n s of a l t e r n a t i v e theories have been reported to date. Of the ' m i s p e r c e p t i o n - o f - p r o b a b i l i t y ' theor ie s , Thaler (1980) appl ied prospect theory to account for several puzzles i n consumer behavior . Shefr in and Statman (1984) p a r t i a l l y appl ied prospect theory to model inves tors ' preference for cash dividends over stock d iv idends . Among the theories of general preference f u n c t i o n a l s , impl i ca t ions of both weighted u t i l i t y and Fre*chet d i f f e rent iab le preference func t iona l - 117 - approach for income i n e q u a l i t y were presented in the respect ive papers of Chew (1983) and Machina (1982b). Weber (1982) derived for homogeneous weighted u t i l i t y agents a Nash e q u i l i b r i u m bidding strategy that i s compa- t i b l e with the 'd i s crepanc ie s ' i n the observed bids under the Dutch auc- t i o n and the f i r s t - p r i c e sea led-bid auct ion reported i n the experiments conducted by Cox, Roberson and Smith (1982). Machina (1982a; 1982b) appl i ed Fre*chet d i f f e rent iab le u t i l i t y theory to obtain condit ions for comparative and decreasing r i s k avers ion , as wel l as for the normali ty of r i s k y - a s s e t demands. Eps te in (1984) appl ied Frdchet d i f f e r e n t i a b l e u t i l i - ty to mean-variance analys i s and provided a re fresh ing and most powerful defense of i t s t h e o r e t i c a l soundness since the scathing attack by Borch (1969) and F e l d s t e i n (1969). With the exception of Machina's and Eps te in ' s works, the above inves - t i g a t i o n s are rather fragmentary i n nature . Among them, the studies of She fr in and Statman, Machina, and Eps te in have d i r e c t relevance to f inance . In order to increase our understanding of the a p p l i c a b i l i t y of the numerous a l t e r n a t i v e preference theories to f i n a n c i a l economics, two l i n e s of research appear worthwhile. The f i r s t i s to d i r e c t l y apply a given theory to model s p e c i f i c s i tua t ions i n finance and obtain i m p l i c a - t ions that can be compared to those based on expected u t i l i t y i n the f i n a n c i a l markets. The other i s d i rec ted towards the d e r i v a t i o n of c o n d i - t ions for preference propert ies re levant to finance such as r i s k aversion and the normali ty of r i s k y - a s s e t demands. This essay i s intended towards the l a t t e r . We focus our a t t ent ion on weighted u t i l i t y and contrast i t with expected u t i l i t y and l i n e a r Gateaux u t i l i t y which i s Fre*chet d i f f e r e n t i - - 118 - able when r e s t r i c t e d to a bounded domain. Weighted u t i l i t y and l i n e a r Gateaux u t i l i t y are se lected because, un l ike other proposed a l t e r n a t i v e s , both are a n a l y t i c a l l y t r a c t a b l e . Under expected u t i l i t y , i f two l o t t e r i e s are i n d i f f e r e n t , then when separate ly mixed with a t h i r d l o t t e r y at the same p r o p o r t i o n , the two new compound l o t t e r i e s must also be i n d i f f e r e n t . This implies that the i n d i f - ference curves i n any simplex i n v o l v i n g 3-outcome l o t t e r i e s are p a r a l l e l s t r a i g h t l i n e s . To accommodate A l l a i s - t y p e choice behavior, these two compound l o t t e r i e s must be allowed to l i e on two d i s t i n c t ind i f f erence curves . I n t u i t i v e l y , the most l i b e r a l compromise i s to permit a set of ind i f f erence curves that do not in t er sec t (or t r a n s i t i v i t y w i l l be v i o l a - t ed ) , and w i l l behave i n conformance with the law of the-more-the-better ( i . e . consistent with f i r s t - d e g r e e s tochas t ic dominance). For t echn ica l convenience, we may also require the ind i f f erence curves to be continuous and smooth. I t i s not s u r p r i s i n g that with so l i t t l e s tructure imposed on the preference order ing , r i s k avers ion i n d i f f e r e n t problem contexts might not be equivalent . It i s therefore necessary to speci fy the sense of r i s k avers ion being re f erred to . We def ined, among others , r i s k avers ion i n terms of c o n d i t i o n a l c e r t a i n t y equiva lent , uncondi t ional c e r t a i n t y equiva- l e n t , mean preserving spread, and r i s k y - a s s e t demand. Without spec i fy ing any preference theory, we proved that r i s k avers ion i n the sense of cond i - t i o n a l c e r t a i n t y equivalent and r i s k avers ion i n the sense of mean preser- v ing spread are equivalent as long as the underlying preferences are com- p l e t e , t r a n s i t i v e , continuous i n d i s t r i b u t i o n , and consistent with f i r s t - degree s tochast ic dominance. This was f i r s t showed for f i n i t e l o t t e r i e s - 119 - i n v o l v i n g r a t i o n a l p r o b a b i l i t i e s , then extended to general monetary l o t t e - r i e s . This also holds i n the comparative context, i . e . , agent A is more r i s k averse than agent B i n the sense of c o n d i t i o n a l c e r t a i n t y equivalent i f and only i f A i s more r i s k averse than B in the sense of simple compen- sated spread. We also showed that , regardless of the u t i l i t y theory, A being more r i s k averse than B i n terms of simple compensated spread impl ies that A w i l l demand less of the r i s k y asset i n a world with one safe asset and one r i s k y asset . In expected u t i l i t y , propert ie s of a preference ordering are l a r g e l y captured i n the agent's von Neumann-Morgenstern u t i l i t y funct ion u ( x ) . I f we can i d e n t i f y i t s non-expected u t i l i t y counterpart , the a n a l y t i c a l t r a c - t a b i l i t y of a general u t i l i t y func t iona l w i l l be grea t ly enhanced. For t h i s purpose, we imposed l i n e a r GSteaux d i f f e r e n t i a l s on u t i l i t y f u n c t i o n - a l s and c a l l e d such a func t iona l a l i n e a r Gateaux u t i l i t y . I ts GSteaux d e r i v a t i v e C(x;F) i s termed a l o t t e r y - s p e c i f i c u t i l i t y funct ion (abbrev. LOSUF) which i n many ways plays the ro l e of the von Neumann-Morgenstern u t i l i t y . For ins tance , consistency with the f i r s t - d e g r e e s tochas t ic domi- nance requires an increas ing g loba l r i s k avers ion i n terms of cond i - t i o n a l c e r t a i n t y equivalents and mean preserving spreads i s character ized by a concave £ . Unl ike expected u t i l i t y , however, the concavity of C i s not equivalent to pointwise l o c a l r i s k avers ion . This gap i s welcome be- cause i t can be used to expla in why people purchase insurance and gamble at the same time. I f l i n e a r Gateaux u t i l i t y can resolve major controvers ies under expected u t i l i t y , why should we be in teres ted i n weighted u t i l i t y which i s a s p e c i a l case of l i n e a r GSteaux u t i l i t y ? At l eas t three arguments can be - 120 - made i n response. F i r s t of a l l , l i n e a r Gateaux u t i l i t y i s not axiomatic . I t i s unclear what preference propert ies are embedded i n l i n e a r Ga*teaux u t i l i t y . In contras t , weighted u t i l i t y i s a consequence of s p e c i f i c as- sumptions about preferences , namely completeness, t r a n s i t i v i t y , c o n t i n u i - t y , monotonici ty , and weak s u b s t i t u t i o n . As long as a dec i s i on maker's preferences conform to these axioms, the analys i s v i a weighted u t i l i t y w i l l be v a l i d . Note that the only axiom that departs from expected u t i l i - ty i s the weak s u b s t i t u t i o n . This appears to render weighted u t i l i t y a n a t u r a l replacement for expected u t i l i t y when a nonl inear preference func- t i o n a l i s c a l l e d f o r . Secondly, to generate a l l the ind i f f erence curves i n a simplex of 3- outcome l o t t e r i e s (c f . F igure 1.1), the amount of information needed under l i n e a r Gateaux u t i l i t y might prove insurmountable as any smooth, non in ter - sec t ing ind i f f erence curves are p e r m i s s i b l e . In comparison, weighted u t i - l i t y i s far more e f f i c i e n t . I t only requires the knowledge of one a r b i - t r a r y ind i f f erence curve and the point at which a l l ind i f f erence curves i n t e r s e c t . Of course, th i s also means that there w i l l ex i s t paradoxes that can be explained by l i n e a r Gateaux u t i l i t y but not by weighted u t i l i - t y . Nevertheless , when the problem context does not require g e n e r a l i t y at the l e v e l of l i n e a r Gateaux u t i l i t y , the much greater e f f i c i e n c y of weigh- ted u t i l i t y may appear a t t r a c t i v e . Most important ly , the s p e c i f i c func t iona l form of weighted u t i l i t y allows us to solve e x p l i c i t l y opt imiz ing problems such as p o r t f o l i o s e l ec - t i o n , intertemporal consumption d e c i s i o n , e tc . For instance , some i m p l i - cat ions i n th i s essay are obtainable under weighted u t i l i t y but not under l i n e a r Gateaux u t i l i t y . One i s the observation that , no matter how r i s k - 121 - averse he may be, a weighted u t i l i t y agent, l i k e h is expected u t i l i t y counterpart , w i l l invest a s t i c t l y p o s i t i v e amount i n the r i s k y asset as long as the expected rate of re turn on the r i s k y asset i s s t r i c t l y greater than the safe rate of r e t u r n . A l s o , we are assured that a r i s k averse ( i n the sense of mean preserving spread) weighted u t i l i t y agent's not-worse- than sets are always convex so that h is weighted u t i l i t y i s quasiconcave i n h i s r i s k y - a s s e t demands. Under l i n e a r Gateaux u t i l i t y , th i s need be assumed. Another r e s u l t unique to weighted u t i l i t y i s the necessary and s u f f i - c i ent condi t ion for the r i s k y asset to be a normal good (c f . Theorem WU8). With weighted u t i l i t y , th i s condi t ion i s obtained by f i r s t opt imiz ing the agent's weighted u t i l i t y to y i e l d the f i r s t and second order cond i t ions , then performing conventional comparative s t a t i c s . This approach is not a p p l i c a b l e to l i n e a r Gateaux u t i l i t y without assuming a s p e c i f i c f u n c t i o n - a l form. Even though some comparative s t a t i c s can be c a r r i e d out under l i n e a r Gateaux u t i l i t y v i a path d i f f e r e n t i a t i o n , e x p l i c i t so lut ions are not obtainable without imposing more s t r u c t u r e s . The above discuss ions point out a na tura l d i r e c t i o n for further r e - search. It should be i n t e r e s t i n g to see how market behaviora l i m p l i c a - t ions obtained under expected u t i l i t y i n some s p e c i f i c f i n a n c i a l economic problems such as intertemporal consumption choice , information value and competit ive bidding strategy w i l l be a l t e r e d under weighted u t i l i t y . It i s poss ib le that the r e s u l t s obtained under the expected u t i l i t y hypothe- s i s are s ens i t i ve to agents' preferences . For example, i t i s we l l known that under expected u t i l i t y the Dutch auct ion and the f i r s t - p r i c e sea led- b id auct ion are isomorphic, so are the E n g l i s h auct ion and the second- - 122 - p r i c e sea led-bid a u c t i o n . Nonetheless, Weber (1982) was able to demons- tra te that they might not be perceived as isomorphic by weighted u t i l i t y maximizers with decreas ing, concave weight funct ions . W i l l the second- p r i c e sea led-bid auct ion remain isomorphic to the E n g l i s h auct ion under weighted u t i l i t y ? W i l l the demand-revealing property of the second-price sea led-b id auct ion continue to hold under weighted u t i l i t y ? On the other hand, some r e s u l t s obtained under expected u t i l i t y might prove robust to preference hypotheses. The equivalence between r i s k aver- s ion i n the sense of mean preserving spread and r i s k avers ion i n the sense of c o n d i t i o n a l c e r t a i n t y equiva lent , proved i n th i s essay, i s such an example. Most l i k e l y , in troduc ing weighted u t i l i t y w i l l c a l l for some extent of modi f i ca t ion i n the r e s u l t s obtained under expected u t i l i t y . The necessary and s u f f i c i e n t condi t ion for the normality of r i s k y - a s s e t demand i s such an example. This condi t ion reveals that , when an agent's u t i l i t y funct ion depends on the underly ing d i s t r i b u t i o n , the a t t r i b u t e s of the r i s k he i s fac ing might a f fect h is market choice behavior . 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Z e c k h a u s e r , R. and E . K e e l e r , "Another Type of R i s k A v e r s i o n , " Econometrica 38 (September 1970): 661-665. - 129 - E S S A Y I I C O M P E T I T I V E B I D D I N G A N D I N T E R E S T R A T E F O R M A T I O N I N A N I N F O R M A L F I N A N C I A L M A R K E T - 130 - 0 INTRODUCTION One problem with the economy of a developing country i s i t s c a p i t a l s c a r c i t y , which makes bank loans r e l a t i v e l y inacces s ib l e to most small busi-nessmen and ordinary consumers. As a r e s u l t , a broad c lass of spon- taneously a r i s i n g arrangements for the mutual prov i s i on of c r e d i t and sa- vings i s wide-spread i n the developing world. The forms of the bulk of these informal f i n a n c i a l i n s t i t u t i o n s are of the r o t a t i n g c r e d i t type, which, i n the anthropo log ica l l i t e r a t u r e , are most frequent ly re ferred to as the ' r o t a t i n g c r e d i t a s s o c i a t i o n ' . In a r o t a t i n g c r e d i t a s s o c i a t i o n , members make regular deposits into a pool which i s a v a i l a b l e to s a t i s f y the borrowing needs of i n d i v i d u a l members i n a ro ta t ing manner. The ac- t u a l organ iza t ion , inc lud ing recruitment p o l i c y , s ize of depos i t , and the method of determining the order by which members receive funds, exh ib i t s a remarkable degree of v a r i a t i o n s i n adaptation to l o c a l socio-economic and c u l t u r a l cond i t i ons . Examples range from an as soc ia t ion i n Keta , Ghana ( L i t t l e , 1957), i n which the order of r o t a t i o n was determined by s e n i o r i t y and the s ize of deposit was not f i x e d , to those popular among the Chinese and Japanese where the organizer gets the f i r s t loan i n t e r e s t - f r e e and the subsequent loans are auctioned off to the highest bidder among the p a r t i - c ipants who have not yet rece ived loans (Geertz , 1962). While r o t a t i n g c r e d i t assoc ia t ions with an e x p l i c i t f inance focus are prevalent among the developing countries (given t h e i r r e l a t i v e l y under- developed c a p i t a l markets) , there are other commonly observed non-market, - 131 - expectation-based exchange a c t i v i t i e s that carry a s i g n i f i c a n t r o t a t i n g - c r e d i t component. Ne ighbor l iness , g i f t exchange, poker c lubs , e tc . are obvious examples. Table 0.1 summarizes the c h a r a c t e r i s t i c s of several v a r i a t i o n s of the r o t a t i n g c r e d i t a s soc ia t ion as wel l as ne ighborl iness and g i f t exchange. The various types of the r o t a t i n g c r e d i t a s soc ia t ion d i f f e r mainly i n t h e i r methods of determining the order of r o t a t i o n among members. In the Chinese and Japanese vers ions , members compete for funds by submitting sea l ed-b ids . The A f r i c a n vers ion i s not very i n t e r e s t i n g since i t s r o t a - t i o n a l order i s determined by s e n i o r i t y or other ' s o c i o l o g i c a l ' c r i t e r i a rather than some form of i n t e r e s t rates ( L i t t l e , 1957). The r o t a t i n g c r e - d i t a s soc ia t ion found i n the Middle East i s mainly for the purpose of pur- chasing durable goods such as automobiles (among I s r a e l i s ) and r e f r i g e r a - tors (among Lebanese). The l a s t example of the r o t a t i n g c r e d i t a s s o c i a - t i o n re fers to the b i l a t e r a l pr iva te arrangement popular among some Indian l a b o r e r s . * Under th i s arrangement, a f ixed amount i s a l ternated between two i n d i v i d u a l s at f ixed i n t e r v a l s , often on pay days. In Asian agr icu l ture -based communities, neighbors gather e f f o r t s to accomplish t h e i r seasonal harvest i n r o t a t i o n . In America, during the p ioneer ing days, s i m i l a r arrangements were common for b u i l d i n g houses, f i g h t i n g f i r e s , e t c . The r o t a t i o n a l nature inherent i n g i f t exchange i s p a r t i c u l a r l y c l ear i n the context of wedding g i f t s . Most people begin by saving (g iv ing wed- These examples were provided by the author's colleagues and f r i e n d s . - 132 - Table 0.1: Examples of Rotational Exchange Examples Characteristics Rotating Credit Associations Neighborliness Gift Exchange Chinese Japanese African Middle East Indian Laborer Organization nature e x p l i c i t organizer purpose an informal fi n a n c i a l market a b i l a t e r a l private arrangement t r a d i t i o n t r a d i t i o n yes no no sa t i s f y i n g borrowing or lending needs mutual aids friendship, etc. Membership q u a l i f i c a t i o n relationship mutual selection s e l f - s e l e c t i o n mutual selection multiple b i l a t e r a l b i l a t e r a l m u l t i l i i t e r a l Information ( c r e d i t - worthiness based on) kinship, job immobility, etc. locatlonal immobility status immobility Deposit form sice frequency regular contribution ($ or kind) aid to neighbors g i f t s given out fixed or variable fixed variable fixed variable Withdrawal form size frequency loan ( p a r t i a l l y refund, $ or kind) aid received g i f t s received fixed or variable fixed variable fixed and once per cycle Assignment mechanism precision of rotation by competitive bidding by seni- o r i t y by lot alternating by needs by t r a d i t i o n e.g. weddings precise imprecise ding g i f t s ) towards the ir wedding days when they 'withdraw' the ir 'sav- ings ' and take on a loan of g i f t s that are paid back over the subsequent marriages of other e l i g i b l e members of the 'wedding c lub ' cons i s t ing most- l y of fr iends and r e l a t i v e s . I n t e r e s t i n g l y , membership i n a wedding club i s at l eas t p a r t i a l l y exogenous. Of s i m i l a r nature i s a poker club which serves c e r t a i n s o c i a l functions without an e x p l i c i t organizer . Some arguments can be made to see insurance as a s p e c i a l form of the r o t a t i n g c r e d i t a s soc ia t ion where the insurance company acts as the orga- n i z e r by s e l l i n g p o l i c i e s and processing c la ims . The purpose of the a r - rangement i s of course r i s k sharing for the members and p r o f i t making for the organizer . From the insurance company an i n d i v i d u a l i s associated wi th , we may i n f e r some pr iva te in format ion . For instance , an automobile owner insured by Preferred Risk Mutual Insurance Inc. i n the United States must be a non-drinker and non-smoker (assuming that people t e l l the t ru th when applying for insurance) . On the deposit and withdrawal s ide , insurance p o l i c y holders make f ixed deposits by paying insurance premium p e r i o d i c a l l y . The withdrawal assignment mechanism i s the key d i f ference from the recognized f i n a n c i a l r o t a t i n g c r e d i t a s s o c i a t i o n . F i r s t of a l l , the withdrawal i s prompted by the occurrence of presumably exogenous events which is random i n cases without moral hazards. Consequently, the r o t a t i o n i s imprecise in the sense that a member may never get withdrawals. Secondly, even i f insured hazards do occur and withdrawals are granted, the s ize of withdrawals w i l l i n general depend on the ac tua l losses which again are randomly determined. - 134 - Even though the anthropo log ica l l i t e r a t u r e concerning the ro ta t ing c r e d i t a s soc ia t ion i s about a century's o ld (Geertz , 1962), the associated repeated intertemporal competit ive bidding process has never been the sub- jec t of a r igorous microeconomic study. The focus of th i s essay i s on the intertemporal bidding process i n the Chinese vers ion of the r o t a t i n g c r e - d i t a s s o c i a t i o n . Because of the interdependency between bids across p e r i - ods, the Intertemporal bidding process for the r o t a t i n g c r e d i t a s soc ia t ion i s d i s t i n c t from having repeated auctions independently across time. Ne- v e r t h e l e s s , r e s u l t s i n the competit ive ( s ing le -per iod) bidding l i t e r a t u r e w i l l be of help i n the development of our r e s u l t s . Since the poineering work of Vickrey (1961), there have been numerous s tudies of the resource a l l o c a t i o n r o l e of various forms of auct ion mar- kets (Oren and Wi l l i ams , 1975; Oren and Rothkoph, 1975; Green and L a f f o n t , 1977; Milgrom, 1979; Wi l son , 1979; Coppinger, Smith and T i t u s , 1980; Forsythe and Isaac, 1980; Myerson, 1981; H a r r i s and Raviv , 1981; R i l e y and Samuelson, 1981; Cox, Roberson and Smith, 1982; Cox, Smith and Walker, 1982; Milgrom and Weber, 1982). (The reader i s a lso re ferred to Stark and Rothkopf (1979) and Engelbrecht-Wiggans (1980) for extensive surveys. Cassady (1967) i s a good source of anecdotal h i s t o r i c a l examples.) Many s tudies were concerned with four types of auct ion market forms - the E n g l i s h auc t ion , the Dutch auc t ion , the f i r s t - p r i c e sealed bid auct ion and the second-price sealed bid a u c t i o n . In the E n g l i s h auct ion , pr ices move upwards i n progress ive ly smaller i n t e r v a l s . The purchaser pays the p r i c e that nobody i s w i l l i n g to bid over . In contras t , pr i ces i n the Dutch auct ion move downward. The bidder who stops the downward p r i c e movement purchases the object at that p r i c e . - 135 - Vickrey argued that the E n g l i s h auct ion i s isomorphic to the second-price sealed b id auct ion where the highest bidder pays the pr i ce of the highest re jec ted b i d . This i s supported i n the experimental work of Cox, Roberson and Smith (1982). V i c k r e y ' s other conjecture - the Dutch auct ion i s i s o - morphic to the f i r s t - p r i c e sealed b id auct ion (where the highest bidder pays the pr i ce of h is own bid) - however, i s f a l s i f i e d i n the same e x p e r i - mental study. The organizat ion of the rest of th i s essay i s out l ined below. A more d e t a i l e d d e s c r i p t i o n of the s tructure of the Chinese vers ion r o t a t i n g c r e - d i t a s s o c i a t i o n , c a l l e d ' H u i ' , 2 i s given i n Sect ion 1 where we introduce use fu l terms and notat ions , and describe eight actual cases of H u i . Sect ion 2 contains some pre l iminary analyses of several smal l , hypo- t h e t i c a l Hui (with 2 or 3 members o n l y ) . The main objec t ive i s to inves - t i g a t e i n a pre l iminary way the r a t i o n a l e for the existence of an informal f i n a n c i a l i n s t i t u t i o n amid the more s o p h i s t i c a t e d , western-derived banking system and at the same time f a m i l i a r i z e the reader with the workings of H u i . Sect ion 3 presents the model's assumptions on an i n d i v i d u a l ' s i n - comes, preferences and expectat ions . We also state a d e f i n i t i o n of an agent's intertemporal re servat ion discount vec tor , which i s compatible w i th , but does not depend on, agents' having access to some i n t e r e s t rate The l a b e l ' H u i ' i s used i n both the s ingular and the p l u r a l form. - 136 - i n a formal f i n a n c i a l market, and prove i t s existence and uniqueness. This allows us to der ive , i n Sect ion 4, the i n d i v i d u a l optimal bidding s trategy under the observed f i r s t - p r i c e auct ion (and the hypothet i ca l second-price auction) with the a d d i t i o n a l r e s t r i c t i o n of concavity and t i m e - a d d i t i v i t y on his von Neumann-Morgenstern u t i l i t y f u n c t i o n , and a decreasing marginal outbidden rate ( increas ing marginal outbidding rate) on his subject ive p r o b a b i l i t y d i s t r i b u t i o n of winning at each p e r i o d . We also discuss some comparative s t a t i c s and e f f i c i e n c y impl i ca t ions of the i n d i v i d u a l optimal bidding s trategy . In order to obtain a t rac tab le form for a Nash e q u i l i b r i u m bidding s tra tegy , we, i n Sect ion 5, impose further r e s t r i c t i o n s , inc lud ing r i s k n e u t r a l i t y . This y i e l d s , for each agent, his ex post Hui borrowing and lending in t ere s t r a t e s . These rates depend on the h i s t o r y of the r e a l i z e d winning b i d , inc lud ing the one for the period i n which he wins the auc- t i o n . Weighted by the Nash-equi l ibr ium-induced p r o b a b i l i t y of winning i n each per iod , corresponding ex ante (nondeterminist ic) in teres t rates r e - s u l t . Sect ion 6 describes an a p p l i c a t i o n of the model b u i l t i n Sect ion 5 to a t a c i t c o l l u s i o n among a small group of suppl iers s e l l i n g an i n d i v i - s i b l e commodity to a s ing le buyer ( e .g . the f edera l government). Sect ion 7 concludes th i s essay by suggesting some p o t e n t i a l d i r e c t i o n s for further research . - 137 - 1 THE GENERAL STRUCTURE AND ACTUAL CASES OF HUI 1.1 The General Structure of Hui A Hui cons is t s of an organizer and N voluntary members of h is choice whom he brings together to form an informal market to s a t i s f y t h e i r bor- rowing and lending needs. For operating the market^ and bearing the de- f a u l t r i s k of each member, the organizer receives an i n t e r e s t - f r e e loan of NA, which i s repaid i n N equal insta l lments of A at each of the N subse- quent per iods . The organizer , on the other hand, poses a common r i s k shared by the N members c o l l e c t i v e l y . Consequently, an otherwise m u l t i - l a t e r a l exchange r e l a t i o n i s replaced e f f e c t i v e l y by b i l a t e r a l ones. Let 0 denote the organizer and n (= 1, 2 , . . . , N) denote the p a r t i c i - pant who succeeds i n bidding for the pool at the nth p e r i o d . Let b ^ n be the b i d submi t t ed by p a r t i c i p a n t i at period n, and b^ = max{b^n} be the highest b id submitted at period n. We denote by A the ' s i z e ' of the per- p e r i o d , before-discount (or before-premium) deposit into the poo l . The member's ac tua l payment at each period i s re la ted to A. In a ' d i s - cou n t -b id ' H u i , each member who has already received funds pays A at every The services provided by the organizer include competit ive recruitment and s e l e c t i o n of members, the execution of auct ions , and the c o l l e c t i o n and d e l i v e r y of depos i t s . - 138 - subsequent p e r i o d . At period n, those who have not yet obtained loans pay A - b ^ a p i e c e . In a 'premium-bid' H u i , each member pays A at every period before he succeeds i n bidding for the poo l . Once he wins, say at period n by b i d d i n g b^, he has to pay A+b n at every subsequent p e r i o d . Although both d i s count -b id and premium-bid Hui are observed, the former i s more po- pular among the Chinese while the l a t t e r seems more to the Japanese's l i - k i n g . The cash flow patterns for the organizer and the N members i n both a d i scount -b id Hui and a premium-bid Hui are summarized i n Table 1.1. From Table 1.1, i t i s c l ear that the number of p a r t i c i p a n t s , N, i n a Hui i s a lso the number of periods th i s Hui i s to l a s t . Note that , among the N members i n a d i scount -b id H u i , member 1, who wins the pool NA- ( N - l ) b ^ i n the f i r s t b i d d i n g , i s a pure borrower, whereas member N, who rece ives NA at period N, i s a pure l ender . The other members l i e some- where i n between. In genera l , a member remains a lender u n t i l he receives loans , at which point h is status changes to a borrower. At each p e r i o d , only lenders are e l i g i b l e to b i d . Since the fund a v a i l a b l e at each period must be granted to one member, 4 the number of bidders decreases by one every p e r i o d , l eav ing the l a s t member to c o l l e c t NA at the end without b i d d i n g . Obviously , an a t t r a c t i v e Hui cons is t s of a 'good' mix of borrow- ers and lenders . A Hui formed by a homogeneous group of borrowers w i l l When no bids are submitted (or , equ iva l en t ly , a l l bids are zero) , the winner i s determined by l o t . In the case of t i e - b i d s , e i ther the fund i s shared equal ly (consequently, future repayments are a lso shared e q u a l l y ) , or a second-stage bidding i s conducted to se lect the winner. - 139 - Table 1.1: Cash Flow Patterns of Hui P a r t i c i p a n t s a. Discount -b id Hui per iod 0 1 . . . n N- l N p a r t i c i p a n t 0 NA - A . . . - A . . . - A - A 1 - A A + ( N - l ) ( A - b 1 ) . . . - A . . . - A - A n • - A -(A-b1) • . . . nA+(N-n)(A-b n ) • • . . . - A - A N - l • - A • " ( A - b ^ - ( A - b n ) • • . . . (N-l)A+(A-b N _^) - A N - A " ( A - b ^ - ( A - b n ) - ( A - b ^ ) NA b. Premium-bid Hui per iod 0 1 n . . . N - l N p a r t i c i p a n t 0 NA - A . . . —A . . . - A - A 1 - A NA - ( A + b ^ - ( A + b ^ -(A+Dl) n • - A • — A • • • • NA+zJl]^ • -<A+bn) • -CA+bn) • N - l - A • - A . . . • — A • • • • N A + Z ^ b i • "(A+Vi) N - A - A . • • - A . . . - A N A + E i = l b i - 140 - provide l i t t l e room for trades . S i m i l a r l y , a group of lenders w i l l f ind a Hui to be quite a boredom since no one w i l l be bidding a c t i v e l y . This su- ggests a dual problem to the one treated here — the organ izer ' s problem. An e f f e c t i v e ( i . e . competit ive) organizer presumably maximizes as h is ob- j e c t i v e funct ion the ' s u r p l u s ' generated from members' p a r t i c i p a t i o n . His choice v a r i a b l e cons is t s of the mix of membership i n terms of t h e i r de- grees of borrowing or lending needs. We s h a l l not study the organizer ' s problem i n th i s essay except to point out from time to time his s a l i e n t features such as the ro l e of defaul t r i s k . 1.2 Actual Cases of Hui In November 1983, the f i n a n c i a l sector i n Taiwan was s t a r t l e d by the l a r g e s t - e v e r - s c a l e Hui defaul t i n her h i s t o r y . This occurred i n a small town of 100,000 people named C h i a - L i . A l l e g e d l y , over one thousand people were involved for a t o t a l amount of four b i l l i o n New Taiwan D o l l a r s (NT$) (approximately US$100 m i l l i o n based on the current o f f i c i a l f ixed exchange rate US$1 = NT$40). This incidence has led several l e g i s l a t o r s to urge for governmental r e g u l a t i o n on Hui operat ion i n Taiwan and has prompted at l eas t one survey on Hui s t a t i s t i c s . Q u a l i f y i n g h i s f igures as conservat ive due to sub- j e c t s ' re luctance to revea l t h e i r ac tua l involvement, Wen L i Chung est ima- ted that the t o t a l Hui membership approximates 85% of the i s l a n d ' s popula- t i o n ; the c r e d i t provided by Hui is roughly US$237.5 m i l l i o n per month, or US$2.85 b i l l i o n annual ly , which i s about 21.92% of the i s l a n d ' s na t iona l income (Chao-Ming, 1983). - 141 - In order to f a m i l a r i z e the reader with the workings of H u i , we c o l - l e c t i n Table 1.2 e ight ac tua l examples found i n Taiwan. Hui 1 was formed among the employees of a CPA f irm; i t s organizer was the personnel admi- n i s t o r who co-signed a l l other employees' pay checks. Hui 2 - 8 were formed among the employees of the state-owned Taiwan Power Company (known as T a i Power) which experiences very low turn-over . Their organizers were uniformly sen ior , tenured employees. Each Hui i s character ized by i t s s t a r t i n g and ending time, the predetermined f ixed payment A, the s ize of i t s membership N (excluding the o r g a n i z e r ) , i t s type (d i scount -b id or premium-bid), and the ac tua l winning b i d s . Organiz ing According to the current p r a c t i c e in Taiwan, a prospect ive organizer w i l l draw up and then c i r c u l a t e among p o t e n t i a l p a r t i c i p a n t s a Hui forma- t i o n proposal with proposed s ize of payment (A) and membership (N), date and frequency of meetings ( e .g . every two weeks or every month) and other features such as the minimum amount of b i d s , rounding-off p o l i c y ( e . g . , $901 and $904 bids w i l l be considered as $900 and $905 r e s p e c t i v e l y ) , e t c . A l l in teres ted par t i e s are i n v i t e d to s ign up and suggest a l t e r a t i o n s of terms. Based on the response and suggestions, i n i t i a l l y proposed terms may be r e v i s e d . When terms and memberships are f i n a l i z e d , a form conta in - ing the agreed-upon terms and the names of members w i l l be d i s t r i b u t e d to a l l members. Usua l ly the form i s designed with space to f i l l i n the win- ning b id and the amount of the r e s u l t i n g pool at each p e r i o d . The data i n Table 1.2 are taken from such forms. Because the i n d i v i d u a l s who provided these forms stopped recording the winning bids a f ter they obtained funds, - 142 - Table 1.2: Ac tua l Cases of Hui Hui s t a r t end N type min b time* Oct . 75 June 77 1,000* 20 d i s c . 300 J u l y 77 Mar. 79 1,000 20 d i s c . J u l y 77 Jan . 79 2,000 18 d i s c . Mar. 78 Feb. 80 2,000 23 prem. Oct . 79 A p r . 82 2,000 30 prem. J u l y 80 June 84 5,000 47 di sc . J a n . 81 May 83 5,000 28 d i s c . 600 J u l y 81 Mar. 83 5,000 25 prem. 900 winning bids 0 1 450 120 200 400 460 1,530 900 2 470 140 210 350 500 1,680 960 3 401 150 200 360 600 1,680 920 4 500 155 150 360 680 1,720 920 5 300** 150 160 360 600 1,610 950 6 135 170 320 550 1,680 960 7 125 170 310 530 1,590 980 8 110 150 340 500 1,750 1,000 9 110 160 330 560 1,610 10 100 150 310 500 1,620 11 115 160 280 500 1,640 12 125 150 280 490 1,520 13 140 160 230 510 1,660 14 130 170 250 600 1,670 15 100 160 250 500 16 50 120 220 430 17 110 210 450 18 210 400 19 350 20 400 21 400 22 420 23 450 * A l l Hui are on monthly b a s i s . * A l l amounts are i n New Taiwan Do l lar s (NT$); US$1 « NT$40. ** The winner was determined by l o t due to absence of b i d s . - 143 - the data are incomplete. Although th i s form i s informal ( e . g . , i t i s not n o t a r i z e d , not signed by the organizer and the members, e t c . ) and i s provided mainly for the r e - cording convenience of the p a r t i c i p a n t s , i t i s , according to a recent court r u l i n g , acceptable as evidence for the existence of f i n a n c i a l claims and l i a b i l i t i e s among Hui p a r t i c i p a n t s (Chao-Ming, 1983). Organizer In return for h i s services i n c l u d i n g r e c r u i t i n g members, conducting auct ions , c o l l e c t i n g payments from each member, d e l i v e r i n g the pooled pro - ceeds to the winner, and most importantly assuming the defaul t r i s k posed by a l l members, the organizer obtains an i n t e r e s t - f r e e loan at the s tar t of the H u i . Should any member d e f a u l t , the organizer must take over the d e f a u l t i n g member's share and the Hui w i l l continue without i n t e r r u p t i o n . To be an acceptable organ izer , one must be be l ieved to be trustworthy and f i n a n c i a l l y capable of assuming the de fau l t ing shares. ' Informal c r e - d i tworth iness ' ( i n the sense that h i s gain from defau l t ing from his o b l i - gations once w i l l be more than outweighed by the loss from a l l future Hui and other informal transact ions ) i s a necessary a t t r i b u t e of any Hui orga- n i z e r at the time a Hui i s formed. We have yet to know a case i n which an i n d i v i d u a l i s able to organize a Hui (or to p a r t i c i p a t e as a member) i n a c i r c l e where he i s known to have defaulted before e i ther as a Hui organ i - zer or a Hui member. Such enforcement by the d i s c i p l i n e of continuous deal ings depends of course c r i t i c a l l y on some general ized d e f i n i t i o n of immobil i ty of which formal c o l l a t e r a l i z a t i o n i s an example. As such, organizers need not be wealthy. The organizers i n our 8 cases were e i ther i n a p o s i t i o n to deter d e f a u l t s , o r , i f defaul t did - 144 - occur, had the a b i l i t y to assume the l o s s . For example, the organizer of Hui 1 had access to a l l members' paychecks. As to Hui 2 - 8 , the l i k e l i - hood of the organizers ' running away i s rather s l im as a job with T a i Power i s considered more valuable than the pool i n most cases. Besides the organ izer ' s character and credi tworth iness , the r i s k of a Hui depends to a large extent on i t s o r i e n t a t i o n and the mix of i t s mem- bersh ip . For instance , because a l l Hui i n Table 1.2 were formed mainly for the saving purpose and involved no businessmen, they were v i r t u a l l y of no defaul t r i s k . By now, a l l the 8 Hui have ended without d i spute . According to our observat ion , the r i s k l e v e l of a Hui increases i f the organizer and/or any members are small businessmen. This i s because i n many instances the businessman members, with higher opportunity costs of c a p i t a l , would b id higher and draw funds from the pool f i r s t while t h e i r a b i l i t y of paying up t h e i r shares depends on the subsequent success of t h e i r business or t h e i r a b i l i t y to borrow from other H u i . It i s worth noting that the recent boom of Hui i n Taiwan has i n t r o - duced the s o - c a l l e d 'pro fe s s iona l organizers ' who have p r o f i t e d from t h e i r entrepreneurship i n a r b i t r a g i n g across Hui or channel l ing funds to l u c r a - t i v e ventures . This new breed of profess ionals usua l ly have good and broad connections with f r i e n d s , r e l a t i v e s , co l leagues , ex-co l leagues , neighbors , e t c . , and consequently have s p e c i a l access to va luable informa- t i o n about e i ther p r o f i t a b l e investment opportuni t ies or people's c r e d i t - worthiness . Being a pro fe s s iona l organizer however does not necess i tate f u l l - t i m e involvement. It i s an occupation often taken up by housewives. To minimize h is r i s k , a prudent organizer must be s e l e c t i v e and i s of ten re luc tant to accept people whom he is not fami lar with as members. - 145 - In immigrant Chinese communities i n North America, a new immigrant seeking membership is often required to e i ther produce a guarantor acceptable to the organizer , or be a saver i n the i n i t i a l periods to e s t a b l i s h his a b i - l i t y to make per iod i c payments ( L i g h t , 1972). In A f r i c a and Middle East where l o t t e r y i s the a l l o c a t i o n a l mechanism for r o t a t i n g c r e d i t a s soc ia - t i o n s , i t i s a common p r a c t i c e that new faces are not allowed to draw l o t s and have to be the l a s t ones to withdraw funds u n t i l they are better known and have es tabl i shed t h e i r c r e d i t a b i l i t y (Ardener, 1964). Members Hui has reportedly exis ted among the Chinese for at l east 800 years (Geertz , 1962) . 5 In a sense, the presence of th i s informal f i n a n c i a l i n s - t i t u t i o n r e f l e c t s a more formal aspect of the e x i s t i n g and genera l ly immo- b i l e r e l a t i o n s h i p s among the members. Today, most sav ing-or iented Hui (which are viewed as more conservat ive but safer) are s t i l l formed among i n d i v i d u a l s who know each other f a i r l y we l l e i ther d i r e c t l y or i n d i r e c t l y . N a t u r a l l y , some people are w i l l i n g to take r i s k for higher returns by j o i n i n g Hui that have greater involvement with businessmen. With some ' s u p e r i o r ' information and by c a r e f u l search and other measures , 6 i t i s not uncommon for one to r e a l i z e a 30% - 40% annualized rate of re turn without much r i s k . b In e a r l i e r H u i , e s p e c i a l l y those found i n agr icu l ture -based communities, the exchange commodity was often in k i n d , being r i c e i n many cases. Today, money i s the only known currency traded i n Hui prevalent i n Taiwan. b For instance , p ick a Hui whose organizer i s your next door neighbor who runs a TV shop and has enough TV sets i n stock which you can lay your hand on i n time i f s i t u a t i o n c a l l s for such auct ions . - 146 - Frequency of Meeting (Bidding) The frequency of Hui meeting and bidding depends to a large extent on the s ize of the payment (A) , the s ize of the membership (N) and the f i n a n - c i a l background of the members. Most sav ing-or iented , middle-c lass -based Hui such as the eight cases we presented are on monthly b a s i s . Bi-weekly meeting i s a lso quite common. I t i s not iced that speculat ive Hui tend to have shorter i n t e r v a l s . Many Hui involved i n the C h a i - L i scandal were a l l eged ly on a d a i l y or b i - d a i l y b a s i s . S ize of Membership The s ize of membership in Table 1.2 ranges from 18 to 47. A member- ship of 47 i s uncommon for a sav ing-or iented , monthly H u i , e s p e c i a l l y given the s ize of payment NT$5,000. 7 In general , the s ize of membership, the s ize of payment and the frequency of meeting are interdependent. I t i s be l ieved that the longer a Hui l a s t s , the greater i s i t s r i s k of d e f a u l t . Most people tend to prefer a durat ion of one and h a l f years to two years so that the pool NA i s not too small and yet the defaul t r i s k i s not unaffordable . I t i s quite common that more than one i n d i v i d u a l share one membership or one i n d i v i d u a l assumes more than one share. The l a t t e r case i s of par - t i c u l a r in t ere s t as i t i s tantamount to permitt ing c o a l i t i o n among b i d - ders . Such prac t i ces al low Hui to d i sp lay a somewhat greater degree of 7 A co l lege graduate's s t a r t i n g monthly sa lary was about NT$12,000 i n 1980. - 147 - f l e x i b i l i t y i n accommodating more des irab le s ize of saving and borrowing. This seems to p a r a l l e l the insurance market where an i n d i v i d u a l chooses among a f ixed menu of p o l i c i e s rather than speci fy the s ize of h is own needs together with the pr i ce he i s w i l l i n g to pay (Rothschi ld & S t i g l i t z , 1976). A member i s also allowed to s e l l h is share to other members or some outs iders before the Hui ends as long as he i s able to obta in the approval from the organizer . This can also be done i f the o r i g i n a l member guaran- tees the creditworthiness of the member(s) he introduces . Discount-Bid vs. Premium-Bid The bulk of Hui i n Taiwan are of the d i scount -b id type. Most Hui found among Chinese in North America, on the contrary , are premium-bid ones. Although these two types are s i m i l a r i n substance, there are i n s t i - t u t i o n a l d i f f erences : a . The payment made by a member who has not withdrawn funds i s A i n a premium-bid Hui and A minus the current winning bid i n a d i s count -b id H u i . This provides an incent ive for a member to take a more ac t ive part i n b idd ing . As a r e s u l t , a d i scount -b id i s l i k e l y to encourage greater p a r t i c i p a t i o n . This i s cons is tent with the general impression that d i scount -b id Hui are more ' e x c i t i n g ' . b . The i n t e r e s t - f r e e loan an organizer obtains i s the same (NA) i n both types of H u i . In the event a member de fau l t s , the per -per iod amount the organizer would be responsible for however i s d i f f e r e n t , being A in a d i scount -b id Hui and A plus the de fau l t ing member's winning bid i n a premium-bid H u i . Moreover, bids i n a d i scount -b id Hui are bounded from below by s tructure ( i t cannot go beyond A ) , but may i n p r i n c i p l e be - 148 - very large i n a premium-bid H u i . The organizer therefore has reasons to prefer d i scount -b id Hui over premium-bid ones. This can at l east p a r t l y expla in the fact that most ' specu la t ive ' Hui are of the d i s - count-b id type. c. The p o o l a v a i l a b l e at p e r i o d n i s NA+E^_^b_^ i n a premium-bid Hui and N A - ( N - n ) b n i n a d i s c o u n t - b i d H u i . The former appeals to people who wish to obtain at least NA when they win. The l a t t e r i s preferred by those who wish to pay l e s s . This argument i s at l eas t to some extent s u p e r f i c i a l s ince , before j o i n i n g a H u i , a p a r t i c i p a n t can se lect a Hui of the c h a r a c t e r i s t i c s that su i t h is preference. d . In a soc iety with wide-spread i l l i t e r a c y , d i scount -b id arrangements have the a d d i t i o n a l advantage of r e q u i r i n g less record keeping as each p a r t i c i p a n t ' s future payments are independent of past winning b i d s . Bidding On the bidding day (often during lunch time of the pay day) , members wishing to bid would submit t h e i r bids to the organizer . The loan a l l o c a - t i o n a l mechanism for Hui has tended to be v i a f i r s t - p r i c e sea led-bid auc- t ions . U n t i l the recent boom of Hui i n Taiwan, the bidding had been r e l a t i - ve ly 'calm' . For example, many saving-minded members simply did not bo- ther to b i d . A member who could not show up for the bidding often autho- r i z e d another member to b id on his behalf or informed the organizer of h is b id i n advance. Af ter the b i d d i n g , only the winning b id was revea led . Recent ly , Hui b idding has become more compet i t ive . The fo l lowing phenomena at tes t to i t : a . The information about needs for loans is guarded as top personal secret - 149 - to prevent s t r a t e g i c compet i t ion. b . E l i g i b l e bidders not i n need of funds would s t r a t e g i c a l l y give fa l se s igna l s (usual ly by t a l k i n g casua l ly about the amount he intends to bid) to induce higher bids from r i v a l b idders . c . Members would withhold t h e i r bids u n t i l the bidding meeting to ensure that t h e i r fr iends or the organizer cannot leak the in format ion . d . Increas ing ly , organizers announce a l l submitted bids without revea l ing the i d e n t i t y of the bidders (except the winner) . This prac t i ce may have impl i ca t ions for the ro l e of information and l earn ing i n the Hui s e t t i n g . It should however be noted that , i n less-commercial ized Hui such as those reported i n Table 1.2, some non-market economic f a c t o r s , e .g . f r i - endship and s o c i a l norms, s t i l l play a r o l e . For instance , i f a member needs funds to h o s p i t a l i z e h is a i l i n g parent, i t i s very l i k e l y that a l l e l i g i b l e bidders w i l l agree upon a low, nominal bid and e f f e c t i v e l y grant a subsidized loan to him (provided of course that he i s not too unpopu- l a r ) . This corresponds to the insurance funct ion of a Hui and other i n - formal market mechanisms. Bids What factors a f fec t the l e v e l of bids? I n t u i t i v e l y , we expect bids to increase i n A and N as suggested by the data i n Table 1.2. A l s o , p r e - mium-bids are expected to be higher than d iscount-b ids s ince the s ize of loan i n the former i s l a r g e r . The economic determinants of a Hui member's bid include at l eas t h is investment opportunity cost of c a p i t a l , h is s tochast ic or nonstochast ic non-investment needs for funds, and a s t r a t e g i c component inherent i n most - 150 - game behavior . F i r s t , consider the investment cost of c a p i t a l i n the formal f i n a n - c i a l markets. In th i s essay, a f i n a n c i a l i n s t i t u t i o n i s considered 'form- a l ' i f i t r e l i e s on e x t e r n a l l y recognized and c o l l a t e r a l i z e d evidence to enforce non-default from without ( e .g . the l e g a l system). In contras t , Hui i s ' i n f o r m a l ' in making use of creditworthiness information generated w i t h in the ( informal) i n s t i t u t i o n to enforce non-default ( i . e . enforcement from w i t h i n ) . The most f a m i l i a r formal f i n a n c i a l market i s the conven- t i o n a l banking system. Almost everybody can save with banks. Therefore the relevant opportunity cost of c a p i t a l for Hui p a r t i c i p a n t s who do not have other investment opportuni t ies would be the bank in teres t rate for saving which i s the same for most i n d i v i d u a l s . For those people who have other investment o p p o r t u n i t i e s , the cost of c a p i t a l i n the formal f i n a n c i a l market i s the bank lending in teres t rate a p p l i c a b l e to him. The r e a l i t y i s however more complicated due to the im- per fec t i on of the formal f i n a n c i a l market. F i r s t of a l l , c a p i t a l r a t i o n - ing does e x i s t . Banks, which cannot demand more than the regulated i n t e r - est r a t e , prefer to deal with large corporat ions due to r i s k cons iderat ion and economy of s c a l e . Loans for small businesses and consumption are a v a i l a b l e , but the process could be forb idd ing ly c o s t l y and the r e q u i r e - ments d i f f i c u l t to f u l f i l l . For example, a standard requirement for small business loan i s two or more noncorporate guarantors. The in t ere s t rate for an unobtainable loan i s e f f e c t i v e l y I n f i n i t y . It i s then not s u r p r i s - ing that many small businessmen of fer a rate as high as 50% annually for - 151 - loans from H u i . For people who p a r t i c i p a t e in more than one Hui (which i s a common p r a c t i c e ) , the i n t e r e s t rate of other Hui might be the relevant opportuni - ty cost of c a p i t a l . This i s e s p e c i a l l y true for the pro fe s s iona l Hui a r - b i t r a g e r s . I t should however be noted that for an informal f i n a n c i a l mechanism such as Hui to sus ta in over a long period of time, the pooled c a p i t a l must be eventual ly channelled to economically productive a c t i v i t i e s which y i e l d r e a l re turns . In other words, the 'rate of r e t u r n ' y on Hui must be sup- ported by growth i n r e a l economic a c t i v i t i e s outside of the Hui system. I f the funds keep c i r c u l a t i n g wi th in a Hui system g iv ing high returns and never flow to the r e a l economic production sector , then th i s Hui system would eventual ly lead to a pyramid. This i s evident i n l i g h t of the 1983 C h i a - L i f i a s c o , i n which the most s t r i k i n g and devastat ing feature of the Hui involved i s a widespread prac t i ce dubbed ' f e e d - H u i - w i t h - H u i ' , i . e . , a p a r t i c i p a n t draws funds from a Hui to make payment i n another H u i . Feed- Hui -wi th -Hui operat ion was also blamed for the f a i l u r e of Chit Fund Corporations i n Singapore during 1972 and 1973 (Chua, 1981). s I r o n i c a l l y , the popular i ty of Hui has rendered an i n d i v i d u a l ' s a b i l i t y to f inance through Hui a s i g n a l of his c r e d i t a b i l i t y . How can a person who cannot be accepted into a Hui expect someone e lse to guarantee h is loan? But i f one can borrow from H u i , why should he need to borrow from the bank? y The 'rate of r e t u r n ' from Hui suggests a poss ib le general e q u i l i b r i u m model which i s beyond the scope of th i s essay. - 152 - As to the s tochast ic or non-stochast ic non-investment needs for funds, the s ing le most important use of loans from Hui has been house pur- chase and r e n o v a t i o n . i 0 Purchases of durable goods, c h i l d r e n ' s education expenses, wedding expenses and fore ign t r a v e l are other uses of funds from H u i . I t seems reasonable to say that Hui p a r t i c i p a n t s have a ' re servat ion p r i c e ' for loans a v a i l a b l e at each period l a r g e l y determined by his oppor- t u n i t y cost of c a p i t a l which depends on (a) h is investment opportunity elsewhere, (b) h is cost of c a p i t a l i n the formal f i n a n c i a l market or other f i n a n c i a l sources, and (c) his personal needs for funds. This i s cons i s - tent with the way Hui p a r t i c i p a n t s c a l c u l a t e the upper bound of t h e i r b i d s . U s u a l l y , an Ind iv idua l whose only a l t e r n a t i v e i s saving with banks w i l l use the bank saving i n t e r e s t rate to c a l c u l a t e the highest discount he can af ford to give up. I f he has other use of funds, a premium w i l l be added to the bas ic saving rate and the maximum affordable bid i s c a l c u l a - ted accord ing ly . Later i n Sect ion 3, we w i l l formal ly define th i s r e s e r - v a t i o n p r i c e for loans as the ' r e serva t ion d i s c o u n t ' . Would a Hui member b id his re servat ion discount? Not i n genera l . How his bid deviates from his re serva t ion discount w i l l be considered i n Sect ion 4 where we study the optimal bidding strategy for Hui members. Due to the s a v i n g - o r i e n t a t i o n , the bids i n Table 1.2 are lower than those i n average H u i . For ins tance , the i n t e r n a l rate of re turn of the The required minimum downpayment i s often more than 60% of the p r i c e . U s u a l l y , i t w i l l take more than one Hui to obtain enough funds for th i s purpose. - 153 - f i r s t winner of Hui 6 i s approximately 1.33% per month, whereas i t i s not uncommon to have 30% - 50% annualized ex post Hui borrowing in t ere s t rate (to be defined i n Sect ion 5, D e f i n i t i o n 5.1) i n many Hui where small b u s i - nessmen are invo lved . The l a t t e r of course have higher defaul t r i s k . Whether the perceived default r i s k leads to higher bids or the other way around has yet to be i n v e s t i g a t e d . Note that not any two Hui on Table 1.2 are s t r i c t l y comparable. For example, Hui 1 and 2 had the same A and N and were both of the d i scount- b i d type, but took place two years apart . As discussed above, bids are to a large extent determined by i n t e r e s t rates i n the formal f i n a n c i a l mar- ke t , which vary over time. Even i f they had existed contemporari ly , mar- ket segmentation might s t i l l prevent bids from a t t a i n i n g p a r i t y . Even though Hui 3 was smaller than Hui 2 by two members, we would ex- pect the winning bids i n Hui 3 to approximately double those of Hui 2. The fact that the bids i n Hui 3 were much lower than expected could be be- cause of the existence of the organizer . The organizer i s expensive to keep and the fewer members a Hui has, the more c o s t l y the organizer i s to each member, everything e lse being constant . We w i l l show i n Sect ion 2 that a Hui with only two members can not a f ford to have an organ izer . I f a l l members' expectations and opportunity in t ere s t rates remain constant over time, one would expect t h e i r bids on the average to be i n - creas ing over time. This i s mainly due to the decreasing number of e l i - g i b l e b i d d e r s who (with payment A - b n ) w i l l enjoy the d i scount . Although none of the winning bid streams i n Table 1.2 are monotone over time, i t could be the r e s u l t of b idders ' changing expectations or opportunity r a t e s . - 154 - Learning As long as a member knows or i s able to estimate his opportunity cost of c a p i t a l , d e r i v i n g the reservat ion discount i s s t ra ight forward . The spread between his re serva t ion discount and his bid is a more s t r a t e g i c i s s u e . The d i scuss ion under Bidding t e l l s us that Hui p a r t i c i p a n t s do be- have s t r a t e g i c a l l y , inc lud ing a c t i v e l y seeking information on r i v a l s ' r e - serva t ion d iscounts . Obviously , other b idders ' opportunity cost of c a p i - t a l , r i s k a t t i t u d e s , bidding s t r a t e g i e s , e t c . are valuable in format ion . A question of i n t e r e s t here i s : Do p a r t i c i p a n t s abstract use fu l i n - formation from the d i s t r i b u t i o n of past winning bids? In repeated s i n g l e - per iod -auctions where the same group of bidders compete f o r , say govern- ment defense p r o j e c t s , i t i s evident that bidders l earn about t h e i r r i - v a l s ' reservat ion pr ices or bidding s trateg ies from past biddings (Green and L a f f o n t , 1977; Milgrom, 1979; Myerson, 1981; Milgrom and Weber, 1982). Therefore , a r e a l i s t i c bidding model must al low for b idders ' Bayesian l earning behavior . In the context of H u i , we f e e l , based on the fo l lowing reasons, that the l earning issue i s not as c r i t i c a l as i n repeated s i n g l e - p e r i o d auc- t i o n s . F i r s t of a l l , once a Hui member wins a b idd ing , he drops out the competit ion for the res t of the Hui d u r a t i o n . Secondly, when a Hui bidder loses i n b i d d i n g , he s t i l l gains i n h is dual ro l e as a ' s e l l e r ' of the l o a n . This suggests a weaker incent ive for cos t ly information search. To keep our analys i s simple i n th i s essay, we w i l l assume that Hui b idders ' current bids do not depend on past winning bids or bids i n genera l . - 155 - Default Roughly speaking, there are two types of Hui d e f a u l t , one incurred by the organizer , the other by the member. Most organizer-caused defaul ts are we l l -p lanned . The plot usua l ly goes as fo l lows . The organizer w i l l r e c r u i t as many members as poss ib le and add to the l i s t of p a r t i c i p a n t s a few nonexistent names. At the s t a r t of H u i , he c o l l e c t s NA as the o r g a n i - z e r . At the next few per iods , he submits high bids i n the name of those nonexistent members and obtains loans . A f t e r h is l i s t of nonexistent names i s exhausted, he simply disappears . A Hui member can avoid th i s type of default by i n s i s t i n g on knowing a l l other members. The other type of defaul t r e s u l t s from one or more members' not being able to make t h e i r shares of contr ibut ions a f ter they have drawn funds from the poo l , pos s ib ly due to unfavorable outcomes of t h e i r investments elsewhere. Although the organizer i s i n p r i n c i p l e the only one who as- sumes a l l defaul t r i s k s posed by members, i t i s a fact that a p a r t i c i p a n t does not completely ignore th i s type of default r i s k when he makes the dec i s ion to j o i n a H u i . While we recognize the presence of the defaul t r i s k i n the Hui set - t i n g , we w i l l not i n th i s essay attempt a model formal ly incorporat ing i t . Instead, i t w i l l be assumed that defaul t r i s k i s n e g l i g i b l e . - 156 - 2 THE ECONOMICS OF HUI WITH TWO OR THREE MEMBERS The purpose of th i s sec t ion i s to provide some basic understanding of Hui by performing some pre l iminary economic analyses on several s i m p l i f i e d Hui examples. S p e c i f i c a l l y , we assume that a l l Hui considered here are defaul t r i s k f r e e and of the d i scount -b id type, and that a l l agents have an e x p l i c i t opportunity in t ere s t rate which remain constant throughout the Hui durat ion . Moreover, the Hui are small i n s i z e , with only two or three members, with or without an organizer . Fol lowing the preceding s ec t i on , we use the fo l lowing notat ions: A : the s ize of the p e r - p e r i o d , before-discount deposit into the poo l ; n ( = 1 , 2, 3): the p a r t i c i p a n t who succeeds i n bidding for the pool at the nth per iod; 0 : the organizer who receives an i n t e r e s t - f r e e loan of NA repaid i n N equal ins ta l lments of A at each of the N subsequent periods; b ^ n : the bid submitted by p a r t i c i p a n t i at period n; b = m a x { b _ £ n } : the highest b id submitted at period n; i r^ : member i ' s opportunity in t ere s t r a t e , i = 1, 2, 3. Two-Member Hui Without An Organizer Consider f i r s t a Hui with 2 members and no organ izer . The ex post cash flows for i t s members are - 157 - time 1 2 member 1 A-bx - A 2 " ( A - b 1 ) A Suppose member i ' s o n l y a l t e r n a t i v e i s to save with banks at a rate r ^ . How much should he b id at time 1? I f he i s to rece ive funds at time 1, the proceeds w i l l go to h is bank account. The (gross) re turn a f ter one period must not be less than A, the amount he has to pay at time 2. I f he i s to lend at time 1 and get the money back at time 2, the Hui lending rate must be at l east equal to the bank s a v i n g rate r ^ . Let v^ denote a b id that s a t i s f i e s the above cond i - t i o n s . Then, ( A - v 1 ) ( l + r ± ) = A, which impl ies A r . v - A - J±— = (2.1) i i i The v^ i n (2.1) can be in terpre ted as the ' p r i c e ' for get t ing a loan of A at time 1. Express ion (2.1) says that the p r i c e ( i n time l ' s d o l l a r ) of a loan A for one period i s the present value of the one-period in t ere s t earned on A . L e t ' s c a l l v^ member i ' s re servat ion discount for the loan a v a i l a b l e at time 1. Note that dv. . d r ± (I+rTT^ > °- In f a c t , regardless of the s ize of membership, a Hui member's reservat ion - 158 - discount for funds a v a i l a b l e at each period i s an increas ing funct ion of h i s o p p o r t u n i t y i n t e r e s t r a t e r^ . We t h e r e f o r e have the f o l l o w i n g p r o p o s i t i o n : P r o p o s i t i o n 2 .1: A Hui member's re servat ion discount increases i n h is op- por tun i ty in t ere s t r a t e . Obvious ly , i f the two members have the same opportunity in t ere s t rate and each knows th i s f a c t , then there i s no point to form a H u i . It w i l l be better off for both members to save with the bank to avoid p o s s i b i l i t y of default and r e t a i n the f l e x i b i l i t y of making withdrawals any time they p lease . This argument can e a s i l y be extended to an N-member H u i : P r o p o s i t i o n 2.2: Suppose a group of i n d i v i d u a l s have an i d e n t i c a l opportu- n i t y i n t e r e s t rate which i s a common knowledge shared by every i n d i v i - d u a l . Then, i t i s to everyone's advantage not to form a H u i . What i f r^ > and both r^ and are known to each member? In th i s case, both members know that I t i s easy to see that member 1 has incent ive to lower his cost of Hui borrowing by bidding under v^. On the other hand, member 2 have no incen- t i v e to b i d under and would b id over v^ only i n an attempt to push up member l ' s b i d . I f member 2 wins with a bid between v^ and , both mem- bers are worse-off . To i l l u s t r a t e t h i s , assume A = $100, r^ = 20% and = 10%. A c c o r d i n g l y , v^ = $16.67 and = $9.09. We can then c a l c u l a t e the ex post i n t e r e s t rates and p r o f i t s from Hui for both members depending on the ac tua l winning b i d . v > = v 2 - 159 - Table 2.1: Ex Post Interest Rates and P r o f i t s for A 2-Member Hui — An Example ex post bids member cash flows i n t e r e s t rate p r o f i t * b x = 16.67 > b 2 b1 = 12.00 > b 2 bl = 9.09 > b 2 +83.33 -100 20% (borrow) -83.33 +100 20% (lend) +88 -100 13.6% (borrow) -88 +100 13.6% (lend) +90.91 -100 10% (borrow) -90.91 +100 10% (lend) 0 8.34 5.6 3.2 9.09 0 bl < b 2 = 9.09 b1 < b 2 = 12 b1 < b2 = 16.67 -90.91 +100 10% (lend) +90.91 -100 10% (borrow) -88 +100 13.6% (lend) +88 -100 13.6% (borrow) -83.33 +100 20% (lend) -9.09 0 -5 .6 -3 .2 0 +83.33 -100 20% (borrow) -8.34 The p r o f i t i s (A-b) ( l+r^) -A for the borrower and A-(A-b)(1+r^) for the lender , where b = max{b^,b 2}. - 160 - It fol lows from the analys i s i n Table 2.1 that: P r o p o s i t i o n 2.3: For a 2-member Hui with r^ > r^, i t i s not Pareto-opt imal for member 2 to win the only bidding at time 1. The actual s p l i t of the p o t e n t i a l p r o f i t depends on these two mem- bers' r e l a t i v e bargaining p o s i t i o n s . I f each member knows only the d i s t r i b u t i o n of h is r i v a l ' s re servat ion d i scount , the bidding strategy becomes much more complicated and w i l l be examined i n Sections 3 - 5 . Two-Member Hui With An Organizer How would the i n t r o d u c t i o n of an organizer change the p ic ture? With an organizer , the cash flows for the p a r t i c i p a n t s become the fo l lowing: p a r t i c i p a n t period 0 1 2 0 2A - A - A 1 - A 2 A - b x - A 2 - A " ( A - b ^ 2A Given the opportunity in t ere s t rate r ^ , member i ( i = 1, 2) w i l l demand at l e a s t a r a t e of r e t u r n equal to r ^ , which means his re servat ion discount v . would be such that l A ( 1 + r i > + o^T " 2 A " v i , which implies v ^ = _ Ar^/ ( l+r^) < 0. Hence, P r o p o s i t i o n 2.4: It i s not poss ib le for a 2-member Hui to support an orga- n i z e r (who obtains an i n t e r e s t - f r e e loan) and yet y i e l d a p o s i t i v e rate of re turn to both members. - 161 - Three-Member Hui Without An Organizer In th i s case, the cash flows are as fo l lows: member time 1 2 3 1 2 ( A - b x ) - A - A 2 " ( A - b ^ 2 A - b 2 - A 3 " ( A - b ^ " ( A - b 2 ) 2A Now, because there w i l l be two b iddings , one at time 1, the other at time 2, each member i w i l l have a reservat ion discount for the funds a v a i l a b l e at time 1 ( i . e . v ^ ) and another for funds a v a i l a b l e at time 2 ( i . e . v ^ ) « I f h i s opportuni ty in t ere s t rate i s r ^ , what w i l l be h i s re serva t ion d i s - counts v ^ and v ^ 2 ? G i v e n h i s f a l l back p o s i t i o n ( i . e . obta ining 2A at time 3) , v ^ and v ^ 2 should s a t i s f y the fo l lowing condi t ions : 2 ( A - v . 1 ) ( l + r i ) 2 = A d + r ^ + A , (2.2) ( A - v . 1 ) ( l + r i ) 2 + A = ( 2 A - v . 2 ) ( l + r . ) , (2.3) ( A - v . ^ d + r ^ 2 + ( A - v 1 2 ) ( l + r ± ) = 2A. (2.4) Re lat ions (2.2) - (2.4) together guarantee a rate of re turn r̂ , and imply V i 2 =I A ^ -TTFH. (2.5) V i l = T A t 2 - l+T- - T l + r T T 2 ^ <2'6> i I Note that v ^ 2 does not depend on v ^ and can be obtained by so lv ing (2.3) and (2 .4 ) . The i n t e r p r e t a t i o n of v^ 2 becomes c l earer i f we take the i n c r e m e n t a l cash f lows of ( - A + v ^ , 2 A - v ^ 2 , - A ) over (-A+v^^ , - A + v ^ 2 , 2 A ) , i . e . , ( 0 , 3 A - 2 v ^ 2 , - 3 A ) . In a sense, 2v^ 2 can be in terpre ted as the pr i ce ( i n the form of pre -pa id i n t e r e s t ) of get t ing an i m p l i c i t loan of 3A at - 162 - time 2, payable at time 3. To v e r i f y t h i s , rewrite (2.5) as (2 .7) : 2 v . 2 = 3 A r . / ( l + r i ) . (2.7) The RHS of (2.7) gives the present value of the one-period in t ere s t on 3A. S i m i l a r l y , (2.6) can be rewri t ten as (2 .8 ) : 3 v n = [ 3 A r i + v . 2 ] / ( l + r 1 ) , (2.8) which of fers a s i m i l a r i n t e r p r e t a t i o n for v ^ . No te t h a t , as s t a t e d i n Propost ion 2 .1 , ^ v ^ / d r ^ > 0 and d v i 2 y ' d r i ^ 0 . Moreover , v^ 2 > v ^ due to the decreasing number of members who would benef i t from the d i scount . We know from P r o p o s i t i o n 2.2 that , i f r^ = r 2 = r^ i s a common know- ledge to a l l members, then there i s no reason for them to form a H u i . Suppose members have heterogeneous opportunity i n t e r e s t ra t e s , and assume w i t h o u t l o s s of genera l i ty that r^ > r 2 > r ^ . P r o p o s i t i o n 2.3 can be ge- n e r a l i z e d to P r o p o s i t i o n 2.5 below: P r o p o s i t i o n 2 .5: Suppose the N members of a Hui have hetergeneous opportu- n i t y i n t e r e s t rates r ^ , i = 1 N, and assume without loss of gene- r a l i t y tha t r^ > r 2 > . . . > r ^ . Then, i t i s to the group's advantage that , at each per iod , the bidder with the highest opportunity in t ere s t rate obtains loan f i r s t . The term 'advantage' i n Propos i t ion 2.5 needs c l a r i f i c a t i o n . For a 2-member H u i , both ' p r o f i t ' and 'ex post i n t e r e s t ra te ' given i n Table 2.1 i s we l l defined because a member is e i ther a pure lender or a pure borrow- e r . I t i s not so for a Hui with 3 or more members due to the ' i l l - b e - haved' cash flows which twice change s igns , consequently might have non- unique i n t e r n a l rates of r e t u r n . - 163 - For a Hui in t ere s t rate to be meaningful, i t i s important that bor- rowing rates be d i s t ingu i shed from lending ra te s . For each member (except the pure lender who obtains funds at the l a s t p e r i o d ) , there are one Hui borrowing rate and one Hui lending r a t e , both derived from the same cash f low s t r e a m . For i n s t a n c e , g i v e n h i s opportuni ty i n t e r e s t rate and cash f low (-A+b^ ,Zk-b^ , - A ) , member 2's Hui borrowing and lending ra te s , denoted by a n d Y^ r e s p e c t i v e l y , are such that ( A - ^ X l + Y j ) - (2A-b 2 ) - ^ and (2A-b 2 ) - ( A - b l ) ( l + r 2 ) = . In other words, to c a l c u l a t e his Hui borrowing r a t e , we assume that he borrows only a f ter he has drawn funds from the pool ; before that , he lends at h is opportunity i n t e r e s t rate r 2 . Symmetrical ly, he lends i n Hui up to the time he obtains funds; therea f t er , he borrows at h is opportunity r a t e . As an example, assume r^ = 20%, r 2 = 16%, r^ = 10% and A = $100. The reservat ion discounts for each member w i l l be as fo l lows: i r i v i l v i 2 1 20% $23.6 $25.0 2 16 19.7 20.7 3 10 13.2 13.6 Suppose bidders shade t h e i r bids under t h e i r re serva t ion discount (This i s an optimal b idding strategy under a set of assumptions. See Sections 3 - 5 . ) so that the ac tua l winning bids are b^ = b 2 = $20. Then, - 164 - the ac tua l cash flows and the Hui borrowing and lending rates for each member are given below: time 20% 15 10 $160 - 80 - 80 -$100 + 180 - 80 -$100 - 100 + 200 16.3% 13.6 16.3% 15.8 This table t e l l s a happy s tory . Member 1 borrowed at 16.3%, lower than his opportunity rate 20%. Member 3 lends at 15.8%, higher than his opportunity rate 10%. Member 2's outcome depends on h i s p o s i t i o n . I f he put the $180 proceeds i n his bank savings account, he is more l i k e l y to see himself as having saved with Hui for one p e r i o d . The in t ere s t rate he earned from lending $80 for one period i s 16.3%, higher than 16%. I f he needed a loan at time 2 for some purpose and considered himself as borrow- ing a one-period loan with a maturity value of $100, h is Hui borrowing rate i s 13.6%, less than 16%. We s h a l l not consider 3-member Hui with an organizer here because i t i s a s p e c i a l case of the model to be studied i n Sections 3 - 5 . - 165 - 3 THE MODEL FOR AN N-MEMBER HUI WITH AN ORGANIZER Notations As in Sections 1 and 2, we w i l l use the fo l lowing notat ions: A : the s ize of the p e r - p e r i o d , before-discount deposit into the pool; n ( = 1 , N): the p a r t i c i p a n t who succeeds i n bidding for the pool at the nth per iod; 0 : the organizer who receives an In teres t - f ree loan of NA repaid i n N equal ins ta l lments of A at each of the N subsequent per iods; b^^: the b id submitted by p a r t i c i p a n t i at period n; b^ : the highest b id submitted at period n; n b : the second highest b id submitted at period n . n Other notations w i l l be defined and explained as they a r i s e . Assumptions Unless otherwise s ta ted , the fo l lowing assumptions are made through- out the paper. Assumption 0: There i s no p o s s i b i l i t y of d e f a u l t . Assumption 1: There i s no Bayesian l earning from past winning bids by i n d i v i d u a l members i n dec id ing t h e i r current b i d s . Assumption 2: Each i n d i v i d u a l i has a de termin i s t i c and known income stream 1^ over the durat ion of Hui p r i o r to the p a r t i c i p a t i o n , where I i = ^ i O ' 1 ! ! I i N ^ ' 1 = 1 » 2 » •**» N " - 166 - Assumption 3: Each i n d i v i d u a l i has a continuous and s t r i c t l y increas ing von Neumann-Morgentern u t i l i t y f u n c t i o n d e f i n e d over h i s income stream. Assumption 4: To each bidder i , the bids from a l l other bidders at period n are drawn independently from p r o b a b i l i t y d i s t r i b u t i o n with support contained i n some i n t e r v a l Tb ,b 1. L - n ' n J To inves t igate an expected u t i l i t y maximizer's optimal bidding s t r a - tegy, we need to define both the ' re serva t ion p r i c e ' i n the context of Hui and ' p o s i t i v e time preference ' : D e f i n i t i o n 3 .1 : The reservat ion discount vector v^ of p a r t i c i p a n t i i s any v e c t o r v^ = ( v ^ » v ^ 2 ' ' ' ' > v ± N - l ^ s u c n t n a t p a r t i c i p a n t i i s i n d i f f e r e n t to the N a l t e r n a t i v e cash flow patterns l i s t e d in Table 3 .1 . In other words, U j ^ Y . ^ ) = U for a l l n = 1, 2, N, (3.1) where Y i l = ( I i 0 - A » I i l + N A - ( N - 1 ) v i l » I i 2 " A ' - " ' I i N " A ) ' Y i n = ( I 1 o - A > I i i - ( A - v i i ) . " - » I l n + N A - C N - n ) v l n » - - - » I i N - A ) » Y i N = ( I i O - A » I i l " ( A - v l l ) » " - ' I l , N - l - ( A - v i , N - l ) » I i N ' W A ) - We s h a l l r e f e r to as p a r t i c i p a n t i ' s r e serva t ion u t i l i t y at the beginning of the H u i . D e f i n i t i o n 3.2: An agent i i s sa id to exh ib i t p o s i t i v e time preference i f h i s p r e f e r e n c e f o r income streams of the f orm ( X Q , . . . ,x^) i s such that - 167 - Table 3.1: Pa r t i c i p a n t i ' s Indi f f e r e n t Cash Flow Patterns i n a Discount-bid Hui period 0 1 •. n • N-l N a l t e r n a t i v e 0 NA -A -A -A -A 1 -A A+(N-l)(A-v i ; L) . • • —A. • -A -A • n -A • " ( A - v i l ) • .. nA+(N-n)(A-v l n) . -A • -A N-l -A • -(A-v 1 ] L) - ( A - v i n ) .. (N-DA+CA-v^j,^) -A N -A - ( A - v ± 1 ) - ( A - v i n ) -< A- vi,N-l> NA - 168 - ( X Q , . . . , X J i x ^ , . . . , x ^ ) i s s t r i c t l y preferred to ( x ^ , . . . , x ^ , . . . ,x^, . . . , x ^ ) i f and only i f x^ > x^ for h = 0, 1, N and k > h. A set of s u f f i c i e n t condit ions for the existence and uniqueness of i s given i n Lemma 3 .1 . Lemma 3 . 1 : The re serva t ion discount vector ex i s t s and i s unique i f the continuous, s t r i c t l y increas ing von Neumann-Morgenstern u t i l i t y e x h i - b i t s p o s i t i v e time preference. Proof: See the Appendix. The proof of Lemma 3.1 involves the cons truct ion of a reservat ion discount vec tor , which i s then shown to be unique. Note that , i n Table 3 . 1 , a l l en tr i e s i n every two adjacent rows, say rows n and n+1, are iden- t i c a l except the nth and the (n+l)th ones. This feature allows the conve- nient backward cons truc t ion of re serva t ion d iscounts , s t a r t i n g with p e r i - ods N and N - l to o b t a i n v^ • From the way v^ i s constructed , v ^ de- pends o n l y on v . , , , which i n t u r n depends on v . , „ , . . . . In other i ,n+1 i ,n+z words, as auctions are conducted and bids revealed at periods 1, 2 n , v ^ f ° r a l l k > n does not change as long as bidder i ' s preference and income stream remain unchanged. Consequently, the property of re servat ion u t i l i t y prev ious ly mentioned also appl ies to the Hui cash flow at any I n - termediate period k. We may c a l l i t member i ' s c o n d i t i o n a l r e serva t ion - k u t i l i t y at period k, denoted by , given some r e a l i z e d cash flow ( C Q , C ^ , . . . , c , , ) . In other words, k-1 U i ( Y i , k + t ) = U ± » fc = °> 1 » • • • » N ~ k « ( 3 ' 2 ) where - 169 - ' Y i , k + t = ( c o ' c i ' - - - j C k - i > I i k " ( A ~ v i k ) ' * " , ^ . k ^ ^ - ^ - ^ . k ^ ^ i . k + t + r ^ - - 0 - ( 3 - 3 ) The c o n d i t i o n a l r e s e r v a t i o n discount vector v, then ref e r s to (v,, , i i k • • • » V J » T i ) » i r r e s p e c t i v e of the r e a l i z e d cash flow (c„,c,,...,c, I, N-1 0 1 k-1 The reservation discount has a f a m i l i a r i n t e r p r e t a t i o n when the i n d i - v i d u a l i s r i s k neutral with time preference being completely described by an i n t e r e s t rate a v a i l a b l e i n some formal f i n a n c i a l sector. In this par- t i c u l a r case, i n d i v i d u a l i ' s reservation discounts represent his market- based opportunity cost of Hui dealing, and can be obtained by equating the present values of the N a l t e r n a t i v e cash flow patterns l i s t e d i n Table 3.1. By tendering bids equal to his reservation discounts at each period, a Hui part i c i p a n t can at least a t t a i n a u t i l i t y l e v e l equal to his reser- vation u t i l i t y , which i s what he can obtain from the formal f i n a n c i a l mar- ket. In th i s sense, reservation discounts describe an i n d i v i d u a l ' s ' f a l l back' po s i t i o n with respect to joining the Hui. It i s clear that, i f a member bids his reservation discounts throughout the duration, his ex post u t i l i t y from p a r t i c i p a t i n g a Hui w i l l never be less than his ex ante re- servation u t i l i t y . When an i n d i v i d u a l has no access to the formal f i n a n c i a l market, or i f the formal f i n a n c i a l market does not exi s t , his reservation discounts can be interpreted as related to some sort of 'as i f i n t e r e s t rate which r e f l e c t s his ' i n t e r n a l ' opportunity cost such as time preference for consumptions. - 170 - 4 O P T I M A L I N D I V I D U A L B I D D I N G S T R A T E G I E S It is shown i n Sect ion 3 that a Hui p a r t i c i p a n t can ensure a u t i l i t y l e v e l , c a l l e d ' r e serva t ion u t i l i t y ' , by submitting a b id equal to h is r e - servat ion discount . In th i s s ec t i on , we consider an expected u t i l i t y ma- x imizer ' s optimal b idding s t r a t e g i e s . To make the problem t r a c t a b l e , we impose, v i a Assumption 3 ' , the requirements of concavi ty , t i m e - a d d i t i v i t y , and p o s i t i v e time preference on i n d i v i d u a l s ' preferences . Assumption 3 ' : The von Neumann-Morgenstern u t i l i t y of each i n d i v i d u a l i , U ^ , i s cont inuous , s t r i c t l y i n c r e a s i n g , concave, and t ime-addi t ive with ( s t r i c t l y ) p o s i t i v e time preference . We can write a t ime-addi t ive von Neumann-Morgenstern u t i l i t y funct ion i n the fo l lowing way: VV-'V = Co^inW' (4'1} I t i s easy to see that s t r i c t l y p o s i t i v e time preference i s , i n th i s case, equivalent to r e q u i r i n g the time preference c o e f f i c i e n t s to be s t r i c t l y decreas ing, i . e . , 1 - X 1 0 > X u > . . . > X 1 N > 0. (4.2) With t ime-addi t ive u t i l i t y , our problem can be solved working back- ward through dynamic programming. From Table 1.1a, we note that , at the l a s t period N, the only member who has not yet received funds obtains NA. There i s no need for bidding and therefore no uncerta inty invo lved . As we proceed backward to periods N - l , N-2, . . . , the number of bidders increases - 171 - by one each time. At each per iod , the prospect of l o s ing e n t a i l s the ex- pected u t i l i t y gain from p a r t i c i p a t i o n i n subsequent per iods . In general , there are N-n+1 bidders at period n, each of whom submits a b id that maxi- mizes his current expected u t i l i t y which incorporates , i n a nested way, the p o t e n t i a l subsequent expected u t i l i t y gain from future per iods . The a c t u a l form of the expected u t i l i t y funct ion depends on the auct ioning me- thod ( i . e . f i r s t - p r i c e or second-pr ice ) . We s h a l l f i r s t consider the case of f i r s t - p r i c e competit ive bidding as the a l l o c a t i o n mechanism. 4 .1 F i r s t - P r i c e Competit ive Bidding In a f i r s t - p r i c e sealed bid Hui where Assumptions 0 (no d e f a u l t ) , 1 (no l e a r n i n g ) , 2 (de termin i s t i c and known pre-Hui income stream), 3' (con- t inuous , s t r i c t l y i n c r e a s i n g , concave, and t ime-addi t ive von Neumann- Morgenstern u t i l i t y with s t r i c t l y p o s i t i v e time preference) and 4 ( inde- pendent and i d e n t i c a l bid d i s t r i b u t i o n for each agent) ho ld , bidder i ' s expected u t i l i t y at period n of a bid b ^ n i s given by expression (4 .3 ) : E U i n < b i n > • WnV^n^-^-^in) + C ^ i n - i ^ i n - ^ ^ ^ i ^ l ^ ^ i n ^ ) ^ " 1 1 ' < 4 * 3 ) i n where E U * f = E u f . . ( b * . . ) = Max Euf (b, . . ) . (4.4) i , n + l i , n + l i , n + l i , n + l v i , n + l The f i r s t term of the RHS of expression (4.3) i s the u t i l i t y for b i d - der i i f he wins at period n, whereas the second term gives h is u t i l i t y i n - 172 - the case of l o s i n g . The component X ^ u ^ I ^ - A + x ) , where x e ( D ^ n > ^ n ] » i s h i s u t i l i t y from p o s t - b i d d i n g income at the current p e r i o d . EU*^ , , i s i , n + l independent of the winning b id at period n and represents the expected u t i l i t y at the next period prov id ing that bidder i submits h is expected u t i l i t y maximizing b id at each of the subsequent per iods . S u f f i c i e n t condit ions for an i n d i v i d u a l ' s bid to maximize the expec- ted u t i l i t y g l o b a l l y are formal ly stated i n Theorem 4.1 below af ter we def ine the terms 'marginal outbidden ra te ' and ' i n c r e a s i n g (decreasing) marginal -outbidden-rate d i s t r i b u t i o n ' , which ensures that the expected u t i l i t y funct ion i s pseudo-concave. D e f i n i t i o n 4 .1: The marginal outbidden ra te for a p r o b a b i l i t y d i s t r i b u t i o n F i s given by F ' / F . An i n t e r p r e t a t i o n of the marginal outbidden rate w i l l be provided shor t ly af ter Theorem 4 .1 . D e f i n i t i o n 4.2: A p r o b a b i l i t y d i s t r i b u t i o n that y i e l d s an increas ing (decreasing) marginal outbidden rate i s c a l l e d an increas ing (decrea- s ing) marginal -outbidden-rate d i s t r i b u t i o n . In Theorem 4 .1 , we suppress the i n d i v i d u a l and time subscr ipts ( i and n) to F , b, I and v without fear of ambiguity. Theorem 4 .1 : Suppose Assumptions 0, 1, 2, 3' and 4 h o l d . Then, for the c l a s s of decreasing marginal -outbidden-rate d i s t r i b u t i o n s , the i n d i v i - d u a l ' s expected u t i l i t y maximizing b id at the nth p e r i o d , b, ex i s t s uniquely; and i s given by the s o l u t i o n to the fo l lowing equation: F ' (b ) = u'(I+NA-(N-n)b) „ F(b) [u(I+NA-(N-n)b)-u(I+NA-(N-n)v)]-[u(I-A+b)-u(I-A+v)+E*] K J with - 173 - E * = d/^^CEU^^-uf 1), (4.6) where EU* i s given i n expression (4.4) and U -n+l i s bidder i ' s cond i - i , n + l t i o n a l reservat ion u t i l i t y at period n+l . Proof: See the Appendix. Equation (4.5) i s the f i r s t order condi t ion for b to maximize the d i f f erence between the expected u t i l i t y and the c o n d i t i o n a l reservat ion u t i l i t y at period n, which, by cons truc t ion , i s a constant . The unique- ness of the expected u t i l i t y maximizing bid i s es tabl i shed by showing that the f i r s t order cond i t ion i s s a t i s f i e d uniquely (s ince the LHS of equation (4.5) i s decreasing by assumption, and the RHS increases i n b by the con- c a v i t y of u ) , and that the second order condi t ion i s s a t i s f i e d l o c a l l y . The decreasing marginal outbidden rate F ' ( b ) / F ( b ) has the i n t e r p r e t a - t i o n as the c o n d i t i o n a l p r o b a b i l i t y that an i n d i v i d u a l b idding b i s 'mar- g i n a l l y ' outbid by his second r i v a l given that he has outbid his f i r s t r i v a l . A decreasing marginal outbidden rate means that as an i n d i v i d u a l r a i s e s his b i d , the increase i n the chance of outbidding his r i v a l must not be slower than the increase i n the chance of being marginal ly o u t b i d . A common c lass of decreasing marginal -outbidden-rate d i s t r i b u t i o n s i s g iven by the power p r o b a b i l i t y d i s t r i b u t i o n , of which the uniform d i s t r i - but ion used i n Sect ion 5 i s a s p e c i a l case. When bids are uniformly d i s t r i b u t e d and bidders are r i s k n e u t r a l , the i n d i v i d u a l expected u t i l i t y maximizing b id w i l l have a closed-form s o l u - s o l u t i o n (given i n expression ( 5 . 3 ) ) . We w i l l have more d e t a i l e d discuss ions on t h i s s p e c i a l case i n Sect ion 5. - 174 - 4.2 Second-Price Competit ive Bidding The formulation of a competitive bidding model for the hypothetical second-price sealed bid Hui i s sim i l a r to that of the f i r s t - p r i c e model, with minor modifications to incorporate the feature that the e f f e c t i v e ( i . e . implemented) discount i s the second highest bid. Under Assumptions 0, 1, 2, 3' and 4, the problem of the expected u t i l i t y maximizer i at pe- rio d n i s stated below: M a X E U i n ( b i n > = -n + ( N - n ) { \ n u i ( I i n - A + b i n ) + E U | S n + 1 } [ F i n ( b i n ) ] N - n - 1 [ l - F i n ( b i n ) ] + (N-n)C J* { X l n u 1 ( I l n - A + y ) + E U ^ n + 1 } d [ F i n ( y ) ] N - n - 1 d F i n ( x ) , (4.7) i n i n where EU* S = EU® ^ ( b * = Max EU® (b. . . ) . (4.8) i,n+l i,n+l i,n+l i,n+l i,n+l The f i r s t term i n expression (4.7) i s the u t i l i t y for bidder i i f he succeeds i n bidding. The second term i s his u t i l i t y i f he f a i l s and turns out to be the second highest bidder. The l a s t term accommodates the case where his bid i s the t h i r d or below. Suppose an i n d i v i d u a l who bids b knows that he has been outbid by one of his r i v a l s . I t i s then to his advantage to be outbid by another r i v a l , so that he, as well as other l o s e r s , can benefit from a higher second bid. P a r a l l e l to the 'marginal outbidden rate' i n the f i r s t - p r i c e case, we need a 'marginal outbidding rate' for our second-price model. D e f i n i t i o n 4.3: The marginal outbidding rate for a p r o b a b i l i t y d i s t r i b u - - 175 - t i o n F i s given by F ' / ( l - F ) . F 1 ( t ) / [ l - F ( t ) ] i s commonly known i n the r e l i a b i l i t y l i t e r a t u r e as the ' f a i l u r e ra te ' (or 'hazard r a t e ' ) , which is the c o n d i t i o n a l p r o b a b i l i t y that a product w i l l f a i l at time t + given that i t has survived up to time t . In the bidding context, the marginal outbidding rate F ' ( b ) / [ l - F ( b ) ] has the i n t e r p r e t a t i o n of the c o n d i t i o n a l p r o b a b i l i t y that an i n d i v i d u a l b idding b w i l l marg ina l ly outbid his second r i v a l given that he has been outbid by his f i r s t r i v a l . To make sure that the expected u t i l i t y func- t i o n i s s ingle-peaked, we require the b id d i s t r i b u t i o n F to y i e l d an i n - creas ing marginal outbidding r a t e . It i s easy to check that the power p r o b a b i l i t y d i s t r i b u t i o n with index greater than or equal to one belongs in th i s category. D e f i n i t i o n 4.4: A p r o b a b i l i t y d i s t r i b u t i o n that y i e l d s an increas ing (de- creasing) marginal outbidding rate i s c a l l e d an increas ing (decreasing) marg ina l -outb idd ing-ra te d i s t r i b u t i o n . A f t e r suppressing the subscr ipts i and n, we state i n Theorem 4.2 a set of s u f f i c i e n t condi t ions for an i n d i v i d u a l ' s bid to maximize h is ex- pected u t i l i t y g l o b a l l y . Theorem 4.2: Suppose Assumptions 0, 1, 2, 3' and 4 h o l d . Then, for the c la s s of increas ing marg ina l -outb idd ing-rate d i s t r i b u t i o n s , the i n d i v i - d u a l ' s expected u t i l i t y maximizing b id at the nth per iod , b, ex i s t s uniquely; and i s given by the s o l u t i o n to the fo l lowing equation: F ' (b ) _ -u'(I-A+b) l - F ( b ) [u(I+NA-(N-n)b)-u(I+NA-(N-n)v)]-[u(I-A+b)-u(I-A+v)+E*] with , (4.9) E * = d / ^ X K U j ^ - u f 1 ) , (4.10) - 176 - where E U * S n + ^ i - s g i v e n i n expression (4.8) and i s h is c o n d i t i o n a l re servat ion u t i l i t y at period n+l . Proof: Omitted. The proof of Theorem 4.2 would be s i m i l a r to that of Theorem 4 .1 . We can f i r s t obtain the f i r s t order condi t ion for a problem equivalent to ( 4 . 7 ) , then show that , with an increas ing marg ina l -outb idding-rate d i s t r i - but ion , the f i r s t order condi t ion i s uniquely s a t i s f i e d by a bid which a l so s a t i s f i e s the second order c o n d i t i o n . 4.3 Impl icat ions In th i s subsect ion, we w i l l discuss several Impl icat ions of Theorems 4.1 and 4.2, making re levant comparisons when appropr ia te . F i r s t of a l l , the fo l lowing c o r o l l a r y t e l l s us that the standard th inking i n the bidding l i t e r a t u r e that bidders w i l l shade t h e i r bids i n f i r s t - p r i c e auctions r e - mains true of agents' behavior i n a H u i . C o r o l l a r y 4 .1 : Under the hypotheses of Theorem 4 .1 , an agent's expected u t i l i t y maximizing bid i s l ess than his re servat ion d i scount . P r o o f : Suppose t h i s i s not t r u e . Then u | > 0 i m p l i e s that the RHS of equation (4.5) i s negat ive . But the LHS of the equation i s p o s i t i v e . This i s a c o n t r a d i c t i o n . Q . E . D . The well-known r e s u l t that second-price auctions are demand r e v e a l - i n g , however, does not in general apply to Hui according to the c o r o l l a r y below. - 177 - C o r o l l a r y 4.2: Under the hypotheses of Theorem 4.2, an expected u t i l i t y maximizer w i l l shade his b id under his reservat ion discount only i f the expected u t i l i t y gain from the next and subsequent periods w i l l more than compensate h is u t i l i t y loss at the current per iod from shading h i s b i d . Proof: Omitted. In a f i r s t - p r i c e sealed bid H u i , an i n d i v i d u a l has no incent ives to b i d i n excess of h is re servat ion discount s ince , when he wins, the amount he obtains i s a decreasing funct ion of h is b i d . I f he lo ses , his b id does not a f fec t what he has to pay. In contras t , due to the d i s p a r i t y between the winning bid and the e f f e c t i v e discount , an i n d i v i d u a l i n a second- p r i c e sealed b id Hui has some incent ive to b id over his re servat ion d i s - count. When a bidder wins, the amount he obtains depends on the highest re jec ted bid rather than his own b i d . When he lo ses , the higher the se- cond b i d , the less he w i l l have to pay. Because there i s a chance that h i s b id might turn out to be the second highest , he has some incent ive to b i d over his r e serva t ion d iscount . He, however, w i l l i n f l a t e his b id only up to a point beyond which his expected u t i l i t y gain from future biddings w i l l outweigh the current bene f i t . I f he perceives s u f f i c i e n t l y high ex- pected u t i l i t y gain from subsequent per iods , he might even bid under his r e serva t ion d iscount . Obvious ly , he w i l l overbid at the l a s t second p e r i - od due to the absence of uncerta inty at the f i n a l p e r i o d . Hence, C o r o l - l a r i e s 4.3 and 4.4 . C o r o l l a r y 4 .3: Under the hypotheses of Theorem 4.1 (Theorem 4 .2 ) , an ex- pected u t i l i t y maximizer i n a f i r s t - (second-) pr i ce sealed b id Hui w i l l lower his b id at the current period i f h is perceived expected u t i l i t y - 178 - gain i n the future increases . C o r o l l a r y 4.4: Under the hypotheses of Theorem 4.2, an expected u t i l i t y maximizer w i l l submit a bid higher than h i s re servat ion discount at the l a s t second p e r i o d . Regardless of the auct ioning methods, poss ib le gains from future b i d - dings tend to dr ive down the present b i d . The poss ib le gain i n the c u r - rent bidding w i l l push the bid even lower i n the f i r s t - p r i c e H u i , but has the opposite e f fec t i n the second-price H u i . Therefore , a member's b id i n the f i r s t - p r i c e Hui i s unambiguously lower than h i s r e s e r v a t i o n d iscount , but th i s i s not n e c e s s a r i l y the case for the second-price H u i . Suppose each b idder ' s re serva t ion discounts c o r r e c t l y measure the true value he places on loans being auctioned of f each p e r i o d . Then, i f a l l bidders fo l low the demand-revealing bidding s trategy , i . e . , b id exact- l y t h e i r re serva t ion d iscounts , the a l l o c a t i o n of loans w i l l be e f f i c i e n t i n the sense that the loan always goes to the most needing agent ( i n d i c a - ted by the highest re serva t ion d i scount ) , l eaving no room for Pareto im- provement. On the contrary , i f someone e lse wins the loan , the most need- ing person can presumably make a side-payment to 'purchase' the loan . An a l l o c a t i o n upon which no Pareto improvement can be made i s c a l l e d a P a r e t o - e f f i c i e n t a l l o c a t i o n of the loan provided wi th in the H u i . I t i s we l l known i n the competit ive bidding l i t e r a t u r e that a f i r s t - p r i c e sealed bid auct ion i s not a P a r e t o - e f f i c i e n t a l l o c a t i o n mechanism while the demand-revealing second-price sealed b id auct ion i s . Do we have p a r a l l e l r e s u l t s in Hui auctions? The answer i s negative (except for the l a s t second period i n a second-price H u i ) . We construct an example to i l l u s t r a t e t h i s . - 179 - Example 4 .1: Suppose ( i ) i n d i v i d u a l s 1 and 2 are competing for a $100 bef ore-discount loan i n a f i r s t - p r i c e H u i , ( i i ) i n d i v i d u a l 1 i s r i s k a v e r s e whereas i n d i v i d u a l 2 i s r i s k n e u t r a l w i th u ( x ^ , x 2 ) = x^+Xx 2, where X = l / ( l + r ) , r i s i n d i v i d u a l 2's non-Hui borrowing as wel l as lending in t ere s t r a t e , ( i i i ) both i n d i v i d u a l s have the same pre-Hui i n - come stream (100,0) , ( i v ) t h e i r respect ive reservat ion d iscounts , ac tua l bids and ex post incomes are given below: i n d i v i d u a l r e serva t ion discount bid ex post income 1 25 12 ( +16,+100) (winner) 2 20 16 (+184,-100) By the d e f i n i t i o n of re serva t ion discounts and the dominance argument, we have (+175,-100) ~ (+25,+100) >- (+16,+100) for i n d i v i d u a l 1, (+184,-100) >- (+180,-100) ~ (+20,+100) for i n d i v i d u a l 2. Suppose i n d i v i d u a l 1 makes a side-payment of $9 to i n d i v i d u a l 2 to 'buy' the loan , the r e s u l t i n g income streams for i n d i v i d u a l s 1 and 2 w i l l be (+175,-100) and (+25,+100), r e s p e c t i v e l y . Since u(+25,+100) = 105 > 104 = u(+184,-100) both i n d i v i d u a l s 1 and 2 are better o f f . S i m i l a r examples can be constructed for a second-price Hui with N > 3. 1 Hence P r o p o s i t i o n 4.1 below: P r o p o s i t i o n 4 .1: The expected u t i l i t y maximizing bids given i n Theorems 4.1 and 4.2 w i l l not i n general y i e l d a P a r e t o - e f f i c i e n t a l l o c a t i o n of c r e d i t s among Hui p a r t i c i p a n t s . This non-Pare to -e f f i c i ency comes from the observation that ne i ther the f i r s t - nor the second-price sealed bid Hui exhib i t s the demand-reveal- - 180 - ing property . The amount by which an i n d i v i d u a l w i l l shade or i n f l a t e his b i d depends on his preferences and expectat ions . When i n d i v i d u a l s ' pre- ferences and expectations are allowed to d i f f e r , we can not i n general expect the highest bidder to have the highest re servat ion d iscount . How- ever, by C o r o l l a r y 4.5 below, we can expect an i n d i v i d u a l to submit a higher bid when his reservat ion discount increases . C o r o l l a r y 4 .5: An i n d i v i d u a l ' s optimal f i r s t - (second-) p r i c e sealed b i d , given i n Theorem 4.1 (Theorem 4.2) increases i n h is c u r r e n t - p e r i o d r e - servat ion d iscount . Proof: See the Appendix. Cox, Roberson and Smith (1982) showed that , when a l l bidders have i d e n t i c a l preferences and expectat ions , the ordinary f i r s t - p r i c e sealed b i d auct ion w i l l be P a r e t o - e f f i c i e n t . C o r o l l a r y 4.5 t e l l s us that th i s i s a lso true for Hui auct ions . Hence, C o r o l l a r y 4.6: C o r o l l a r y 4 .6: Under the hypotheses of Theorem 4.1 (Theorem 4 .2 ) , the f i r s t - (second-) p r i c e sealed b id Hui w i l l be a P a r e t o - e f f i c i e n t a l l o c a - t i o n mechanism for c r e d i t i f a l l i n d i v i d u a l s possess the same tastes ( i . e . u t i l i t y funct ions and time preference c o e f f i c i e n t s ) and expecta- t ions ( i . e . bid d i s t r i b u t i o n s at the current and each of the subsequent p e r i o d s ) . I t was mentioned prev ious ly that the ac tua l outcome of a Hui depends on the composition of i t s members. One question of i n t e r e s t i s : How does a member's time preference af fect h is optimal bid? Other things being equal , a higher preference for current consumption (character ized by a higher time preference c o e f f i c i e n t for the current period) should lead to a higher b id for loans c u r r e n t l y a v a i l a b l e . The r e s u l t that supports th i s - 181 - i n t u i t i o n i s formal ly stated i n C o r o l l a r y 4.7 . C o r o l l a r y 4 .7: Under the hypotheses of Theorem 4.1 (Theorem 4 .2 ) , an i n d i - v i d u a l optimal bid i n a f i r s t - (second-) p r i c e Hui increases as h i s r e - l a t i v e time preference for current consumption increases . Proof: See the Appendix. - 182 - 5 A NASH PROCESS OF INTEREST RATE FORMATION Having derived the i n d i v i d u a l optimal bidding strategy for an agent with a t ime-addi t ive von Neumann-Morgenstern u t i l i t y and studied some of i t s impl ica t ions i n Sect ion 4, we are ready to explore the question of how the winning bids of u t i l i t y maximizing agents w i l l determine for Hui mem- bers t h e i r respect ive i n t e r e s t rates ex ante and ex post . The d e f i n i t i o n s of in t ere s t rates which are appropriate in the Hui context w i l l be given a f t e r the d e r i v a t i o n of the Nash e q u i l i b r i u m i n d i v i d u a l bidding programs which w i l l y i e l d the relevant i n t e r e s t rate i n an endogenous manner. In a d d i t i o n to t r a c t a b i l i t y , the Nash model of s t r a t e g i c i n t e r a c t i o n has found empir i ca l support i n experimental studies of the competit ive bidding behavior for s ing le as we l l as mul t ip l e object auctions (Cox, Roberson and Smith, 1982; Cox, Smith and Walker, 1982). Furthermore, the Nash assump- t i o n may be more p l a u s i b l e given the auct ioning mechanism (sealed bid) used by H u i , which helps members to remain anonymous. In th i s s ec t i on , we w i l l demonstrate that the optimal i n d i v i d u a l b i d - ding s trateg ies derived i n Sect ion 4 w i l l lead to a Nash e q u i l i b r i u m under r i s k n e u t r a l i t y and l i n e a r b id d i s t r i b u t i o n s . The r i s k n e u t r a l i t y assump- t i o n , stated below as Assumption 3", replaces Assumption 3 ' . Assumption 3": Each i n d i v i d u a l i i s r i s k neutra l with the t ime-addi t ive u t i l i t y d e f i n e d over h i s income stream x^ = (X̂Q, . . . , x ^ ) and given by expression (5.1) below: U . ( x . ) = £ N . \ , x , , (5.1) i v i n=0 i n i n ' - 183 - where 1 = X.n > X. , > . . . > X.„ > 0. iO i i iN We have two j u s t i f i c a t i o n s for the r i s k n e u t r a l i t y assumption. F i r s t , i t helps us avoid confounding the r i s k a t t i tude in i n d i v i d u a l s ' b idding behavior which i s the main focus of th i s essay. Second, observa- t ions of the ac tua l d i spers ions of discount bids tend to be r e l a t i v e l y smal l compared with e i ther the loan a v a i l a b l e or the pre-Hui incomes so that we are v i r t u a l l y looking at gambles whose outcomes c l u s t e r wi th in a small i n t e r v a l for which a l i n e a r funct ion i s a good approximation. This seems to accord with observations that Hui p a r t i c i p a n t s tend to use mone- tary va lue , rather than some subject ive u t i l i t y worth, i n t h e i r b id c a l c u - l a t i o n (Huang, 1981). Under r i s k n e u t r a l i t y and uniform bid d i s t r i b u t i o n s , the i n d i v i d u a l opt imal f i r s t - (second-) pr i ce b id given by expression (4.5) (expression (4 .9)) w i l l be l i n e a r i n the reservat ion discount . Since a uniform d i s - t r i b u t i o n i s preserved v i a a l i n e a r transformation, we can replace Assump- t i o n 4 by Assumption 4 ' . Assumption 4 ' : Each i n d i v i d u a l i ' s re servat ion discount for the fund a v a i l a b l e at period n, v ^ n , i s independently drawn from a common uniform d i s t r i b u t i o n H n with support [ Y n » v n l » w n e r e n = 1, N-1; i . e . , H (v. ) = (v. -v ) / (v -v ) (5.2) n i n i n —n n —n for a l l i , where n = 1, N-1. Furthermore, each i n d i v i d u a l i ob- serves h i s own reserva t ion discount v. before he submits h is b id b. . i n i n Assumption 4' impl ies that each i n d i v i d u a l i ' s bid for the fund a v a i l a b l e at the n th p e r i o d , b i n , w i l l be uniformly d i s t r i b u t e d over the i n t e r v a l [b , b . ] , where b and b correspond, through a l i n e a r r e l a - - 184 - t i o n s h i p , to and v , r e s p e c t i v e l y , and are i n d i v i d u a l - s p e c i f i c because d i f f e r e n t bidders might perceive d i f f e r e n t gains from future b iddings . In order to produce consis tent expectat ion, we w i l l assume that each agent be l ieves that every other bidder has the same discounted expected gain from future biddings as h i s own, contingent on lo s ing at th is p e r i o d . Since an agent's expected gain i s discounted using his appropriate time preference c o e f f i c i e n t , we s h a l l re fer to i t as s u b j e c t i v e l y discounted expected gain from future bidding p a r t i c i p a t i o n . This r e s t r i c t i o n i s formal ly stated as Assumption 5: Assumption 5: Each i n d i v i d u a l i be l ieves that a l l other b idders ' subjec- t i v e l y discounted expected gains from future bidding p a r t i c i p a t i o n , con- t ingent on l o s i n g i n the current b idd ing , are equal to h is own. We s h a l l denote member i ' s s u b j e c t i v e l y discounted expected gain at per iod n by G*n- Note that G £ n i s h is expected gains from biddings taking place i n periods n+l, n+2, N-1, s u b j e c t i v e l y discounted to per iod n, and that G | Q can be in terpre ted as member i ' s expected ' surp lus ' for j o i n - ing the H u i . Given the modified assumptions, the Nash equ i l ibr ium bidding programs for the f i r s t - p r i c e and second-price sealed b id Hui are given i n Theorem 5.1 and Theorem 5.2, r e s p e c t i v e l y . To s impl i fy nota t ions , we l e t m = N- n+1 denote the number of e l i g i b l e bidders (those who have not yet received funds) at period n . Theorem 5 .1: Suppose Assumptions 0, 1, 2, 3", 4' and 5 h o l d . Then, the Nash e q u i l i b r i u m bidding program for i n d i v i d u a l i in a f i r s t - p r i c e s e a l - ed bid Hui i s given by - 185 - b*in = V i n - 5TT <Vin-V " 5" G ln> < 5 ' 3 > where G * ^ i s the s u b j e c t i v e l y discounted expected gain from future b i d - ding p a r t i c i p a t i o n , given i t e r a t i v e l y by G l n = ^i.n+l^in^^^W^l^l' ^ V i , n + r b ! , n + l ^ b i , n + r b n + l > m " 2 + / * f \ , ( X - V i , n + l ^ * i ! n + l ^ X - b n + l ) m " 3 d x l ' <5'4> i , n+1 for n = 1, 2, . . . , N - l . Proof: See the Appendix. The proof involves i d e n t i f y i n g the b id funct ion (5.3) as a candidate for a Nash e q u i l i b r i u m and v e r i f y i n g that i t i s indeed the case. Theorem 5.2: Suppose Assumptions 0, 1, 2, 3", 4' and 5 h o l d . Then, the Nash e q u i l i b r i u m bidding program for i n d i v i d u a l i i n a second-price sealed bid Hui i s given by: hf = v. + - r r (v -v, ) - - G * S , (5 .5) i n i n m+1 n i n m i n ' g where G* i s the s u b j e c t i v e l y d i s c o u n t e d expected g a i n from future i n bidding p a r t i c i p a t i o n , given i t e r a t i v e l y by: G * S = (K ) [ (m-2)/(b , . - b , 1 ) m ~ 2 ] * i n i,n+1 i n 1 n+1 -n+1 J { ( m - 2 ) / ^ + 1 ( x - b n + 1 ) m - 3 ( v + 1 - x ) d x -n+1 + ^ / ^ i . n + l ^ t n + l ^ ^ n + l ^ ^ ^ < 5 ' 6 ) i,n+l i,n+l for n = 1, 2, N - l . - 186 - Proof: Omitted s ince i t i s s i m i l a r to the proof of Theorem 5.1 . We summarize i n Table 5.1 the Nash bidding strategy for period n, as w e l l as the corresponding expected values and variances of the e f f e c t i v e discounts ( i . e . winning b id under the f i r s t - p r i c e method and second h igh- est b id under the second-price method). With r i s k n e u t r a l i t y and uniform d i s t r i b u t i o n s , Vickrey (1961) showed tha t , i n a s ing l e -objec t auct ion , the expected pr ices under the f i r s t - and the second-price auct ioning methods are the same, but the corresponding variance i s higher under the l a t t e r . In our Hui bidding model, s i m i l a r r e s u l t s do not appear obvious. I t i s s traightforward to check that , for a l l m > 2, the variance of the e f f e c t i v e discount ( i . e . the b id that i s implemented) i s higher under the second-price than under the f i r s t - p r i c e f f s s a u c t i o n i n g method. I f we f u r t h e r assume that G* = G* = G* = G* for ° i n n n i n a l l i , then the expected e f f ec t ive discount w i l l be higher under the se- cond-price than under the f i r s t - p r i c e method for a l l m. Therefore , from a borrower's s tandpoint , i t seems reasonable to conjecture that the f i r s t - p r i c e method w i l l be preferable to the second-price method. When m = 2, the second-price method, from the l ender ' s point of view, dominates the f i r s t - p r i c e method since the former y i e l d s a higher expected e f f e c t i v e discount with the same var iance . However, given the m u l t i - p e r i o d nature of H u i , the double r o l e played by i t s members, and the fact that the same auct ioning method must be followed c o n s i s t e n t l y throughout a c y c l e , i t i s not present ly c l e a r how to carry th i s l i n e of argument to favor one method over another. Since the auctioned object in Hui i s a homogeneous monetary loan, i t w i l l be des irab le to have some measure by which one can compare the per- - 187 - Table 5.1: Nash Bidding Strategies and Their Der iva t ive s R e s t r i c t i o n s : - no defaul t , no learning - r i s k n e u t r a l i t y with pos i t i ve time preference - l inear d i s t r i b u t i o n of reservat ion discount v over [v,v] - i d e n t i c a l subjec t ive ly discounted expected gains G* F i r s t - P r i c e Second-Price Nash bidding strategy, b v - Jfa (v-v) - ± G* v + ^ (v-v) - i G* b 2 - 5 G * 2 + s i r <*-2> " ¥ G * H v + (v-v) - I G* v - I G* Bid d i s t r i b u t i o n , F(b) ( v - v ) / ( v - v ) ( v - v ) / ( v - v ) E ( b f ) , E ( b s ) v + — S i ! — (v-v) - I G* v - 2 m . (v-v) - 1 G* " (m+1) 2 m (m+1) 2 " m = y + m 2 + l . (v-v) - _- G* (m+1) 2 m V a r ( b f ) , V a r ( b s ) Slf (v-v) 2 2(m-l )m 2 ( - _ y ) 2 (m+1) H(m+2) " ( m + l ) 4 ^ ) m = N-n+1: number of bidders at period n formance of d i f f e r e n t Hui from one's point of in teres t ( e . g . , borrowing or l e n d i n g ) . This measure would preferably be in some form of in teres t r a t e s . One i s at tempted to c o n s i d e r the r a t e r n that solves equation (5.7) below: ^ ~ J ( A - b k ) ( l + r n ) n _ k + Z k f n + 1 A ( l + r n ) n ~ k = NA+(N-n)(A-b f l), (5.7) f s where b . = 0 and b = b (b ) under the f i r s t - (second-) pr i ce auct ioning 0 n n n v method. This measure i m p l i c i t l y assumes an i n d e n t i c a l in t ere s t rate for bor- rowing and l e n d i n g — which m i s l e a d i n g l y looks l i k e some kind of ex post in teres t rate for p a r t i c i p a n t n who won the bidding at period n . This i n t e r n a l rate of re turn however turns out a poor choice given the t y p i c a l Hui cash flows which, except for the organizer and p a r t i c i p a n t N, change signs twice, y i e l d i n g non-unique so lut ions for r ^ . In Sect ion 2, we motivated, using a numerical example, the d i s t i n c - t i o n between Hui borrowing and lending r a t e s . We now formal ly define them. D e f i n i t i o n 5.1: The ex post Hui borrowing i n t e r e s t rate to member n (who won the loan at period n ) , n = 1, . . . , N, with opportunity cost of c a p i - t a l i s the rate y b that solves equation (5.8) below: NA+(N-n)(A-b n ) - £ ^ J ( A - b k ) ( l + r n ) n ~ k = ^ ! ! t t f l A ( l + Y b i ) n ~ \ (5.8) f s where b n = b = 0 and b, = b, (b ) , k = 1 n , under the f i r s t - (second-) p r i c e auct ioning method. D e f i n i t i o n 5.2: The ex post Hui lending i n t e r e s t ra te to member n (who won the loan at period n ) , n = 1, N, with opportunity cost of c a p i t a l - 189 - r i s the rate v that solves equation (5.9) below: n n ^ : J ( A - b k ) ( l + Y ^ ) n _ k = NA+(N-n)(A-b n ) - ^ ! I n + 1 A ( l + r n ) n ~ k , (5.9) f s where b . = b „ = 0 and b. = b. (b, ) , k = 1, . . . . n , under the f i r s t -0 N k k k (second-) pr ice auct ioning method. Note that the ex post Hui in t ere s t rates to the nth member defined above are functions of the winning bids (or the second highest bids) at periods 1, 2, n, and are independent of those at periods n+l, N- 1. Therefore , one member's ex post Hui in teres t rates w i l l in general d i f f e r from another ' s . For an i n d i v i d u a l deciding on whether to j o i n a H u i , an ex ante (ex- pected) in t ere s t rate i s perhaps more re l evant . D e f i n i t i o n 5 .3: Given member i ' s opportunity cost of c a p i t a l r^ and p l a n - ned b i d d i n g program ( b ^ , b ^ 2 > • • • » b ^ J J - 1 ^ ' ^ S e x a n t e Hui borrowing i n t e r e s t rate R b i s given by expression (5.10) below: where " ^ ( b ^ ) , k = 1, N - 1 , i s the p r o b a b i l i t y that member i w i l l win the l o a n at p e r i o d k by s u b m i t t i n g a b i d equa l to b and E ( y b ) solves equation (5 .11): ^ : J [ A - E ( b k ) ] [ l + r 1 ] n " k + ^ n + 1 A [ l + E ( Y b ) ] n " k = NA+(N-n) [ A - E ( b n ) ] (5.11) f o r n = 1, 2, N-1; E ( b n ) i s the expected e f f e c t i v e discount given i n Table 5 .1 , and b Q = 0. D e f i n i t i o n 5.4: Given member i ' s opportunity cost of c a p i t a l r^ and p lan- ned b i d d i n g program ( b ^ , b ^ > • • • >b± N - 1 ) , h i s ex ante H u i l e n d i n g - 190 - Interest rate i s given by expression (5.12) below: R i = €\ tCl^-Wl K< bln> E <tf' < 5 ' 1 2 > where E ( Y n ) solves equation (5.13): ^ ~ J [ A - E ( b k ) ] [ l + E ( Y I J ) ] n " k + ^ n + 1 A [ l + r 1 ] n " k = NA+(N-n) [A-E(b f l ) ] . (5.13) f o r n = 1, 2, N - l ; E ( b n ) and H n ( b i n ) a r e t n e same a s i - n D e f i n i t i o n 5 .3 . Note that the bids that are used to c a l c u l a t e the ex ante in t ere s t rates i n D e f i n i t i o n s 5.3 and 5.4 are expected e f f e c t i v e discounts while those used i n D e f i n i t i o n s 5.1 and 5.2 are r e a l i z e d e f f e c t i v e d iscounts . E s s e n t i a l l y , the ex ante Hui borrowing ( lending) i n t e r e s t rate to a member i s his winning-probabi l i ty -weighted average of the expected borrowing ( lending) i n t e r e s t rates induced by the expected e f f e c t i v e discounts throughout the d u r a t i o n . In general , we would expect d i f f e r e n t members to have d i f f e r e n t ex ante Hui i n t e r e s t rates s ince the ir expectat ions , income streams, as wel l as time preference c h a r a c t e r i s t i c s , may d i f f e r , g i v i n g r i s e to d i f f e r e n t re serva t ion d i scounts . - 191 - 6 AN APPLICATION TO COLLUSION AMONG SEVERAL SELLERS UNDER REPEATED AUCTIONS11 The ro ta t ing c r e d i t a s soc ia t ion studied in previous sect ions can be appl i ed to a form of t a c i t c o l l u s i o n among a small group of s e l l e r s in a se- q u e n t i a l bidding se t t ing with a s ing le agent buying at regular i n t e r v a l s , v i a sealed b id auct ions , an i n d i v i s i b l e commodity from a f ixed group of e l i g i b l e s e l l e r s . The form of conspiracy is non-cooperative i n the sense that once the ru les are agreed upon and fol lowed, the i n d i v i d u a l behavior i s prof i t -maximiz ing i n the usual Nash sense. 6.1 The Structure of Rotating Credit Collusion Consider a buyer buying at regular i n t e r v a l s v i a sea led-bid auctions a s i n g l e i n d i v i s i b l e commodity suppl ied by a s e l l e r from a f ixed group of N s e l l e r s . 1 2 A c t u a l bids submitted are not to exceed the buyer's re servat ion r p r i c e denoted by b . The s e l l e r s enter into bidding at no cos t s . In the absence of any c o l l u s i o n , s e l l e r s maximize t h e i r discounted expected p r o f i t s given the act ions of others . In a c o l l u s i o n with side payments, the s e l l e r with the lowest cost w i l l be pre - se lec ted to win the auct ion at any period and the s p o i l s w i l l be s p l i t according to some pre-agreed sharing r u l e . L i This sect ion i s e s s e n t i a l l y taken from Chew, Mao and Reynolds (1984). 1 2 A s i m i l a r s tructure can be defined for a s ing le s e l l e r and several buyers repeated bidding s e t t i n g . - 192 - In ro ta t ing c r e d i t c o l l u s i o n , the N s e l l e r s agree, p r i o r to the s tar t w of an N - p e r i o d b i d d i n g c y c l e , on a withdrawal bid l e v e l b which i s less r than b . At p e r i o d n during an N-period bidding c y c l e , only the m (= N- n+1) ' l i v i n g ' bidders (those who have yet to win an auct ion in any given w c y c l e of N p e r i o d s ) are a l l owed to b i d at or lower than b . The n - l 'dead' s e l l e r s (those who have won once p r i o r to the given period) wi th- w draw by s u b m i t t i n g b i d s above b , f o r f e i t i n g t h e i r chance to win but g i v i n g an appearnace to the buyer of s t i l l being ac t ive b idders . The i n - cent ive to do th i s i s provided by the knowledge that the game w i l l r e s t a r t a f t er each of the remaining l i v i n g s e l l e r s has won once. The f i n a l 'win- w ner w i l l r e c e i v e b at p e r i o d N. This i s accomplished by submitting a w w b i d of b under f i r s t - p r i c e auctions or submitting a b id at less than b w but r e c e i v i n g b from the next highest bid submitted by a predetermined ( e . g . , the winner at period 1) dead s e l l e r under second-price auct ions . 6.2 Assumptions The fo l lowing notat ion w i l l be used i n th i s s e c t i o n . N : the number of s e l l e r s ; t>in: i t h s e l l e r ' s bid at period n; b^ : the lowest ( i . e . winning) bid at period n; n b s : the second lowest b id at period n; n f s b = b (b ) i n a f i r s t - p r i c e (second-price) s e t t ing ; n n n b : the s ing le buyer's re servat ion p r i c e ; b W : the withdrawal bid l e v e l , b W < b r . - 193 - Other notat ions w i l l be explained when they appear. Assumption 0: There i s no d e f a u l t . Assumption 1: There i s no Bayis ian l earning from past winning b i d s . Assumpt ion C2: Each s e l l e r i has a de termin i s t i c and known costs stream over time: c^ = » c ^ 2 » * * * , c i t ' * * * ^ * Assumption C3: Each s e l l e r maximizes his expected net present value of his p r o f i t s stream given a market discount fac tor p. We d e f i n e below a re serva t ion p r i c e v ^ n for the i t h s e l l e r at period n based on the c e r t a i n knowledge (Assumption 0) that the worst he can do w i s to receive b at the Nth auct ion: p - ( » - " ) ( v l n - C l n ) - b » - c 1 N . (6.1) We w i l l r e f e r to v? = {v. , . . . , v . „ } , n = 1, 2, . . . . N, as the i t h s e l l e r ' s i i n iN — w r e s e r v a t i o n p r i c e vector at period n . Obvious ly , v^ = b for a l l i . Assumption C4: Every s e l l e r at period n bel ieves that the reservat ion p r i c e of the other s e l l e r s are drawn independently from a common uniform d i s t r i b u t i o n over [v ,v ] . L - n ' n J The discounted expected ga in from future p a r t i c i p a t i o n i n b idding for s e l l e r i at p e r i o d n-1 (n = 1, N-1) , denoted by > i s defined by: G, . - p{P, (b, )(b - v , )+[ l -P, (b, )]G, }, (6.2) i , n - l r 1 i n i n n i n 1 i n i n J i n ' and G i , N - l = °> < 6 ' 2 '> where P i n ( b ^ n ) denotes the i t h s e l l e r ' s subject ive p r o b a b i l i t y of winning the a u c t i o n wi th a b id of b ^ n at period n . The p r o b a b i l i t i e s of winning - 194 - can be derived af ter we obtain the Nash e q u i l i b r i u m bidding s trateg ies i n the next subsect ion. In p a r t i c u l a r , G ^ Q represents the o v e r a l l discounted expected gain from p a r t i c i p a t i o n i n the conspiracy at the s tar t of each bidding c y c l e . Assumption C5: Every s e l l e r bel ieves that the discounted expected gain from p a r t i c i p a t i n g i n future biddings for a l l other s e l l e r s , contingent on l o s i n g i n the c u r r e n t p e r i o d , are equal to h is own, i . e . , G . = G for a l l i , where n = 1, . . . , N. Assumptions C4 and C5 are symmetry assumptions making the m (= N-n+1) s e l l e r s more a l i k e . They are imposed i n order to avoid using the t r a d i - t i o n a l homogeneous s e l l e r s assumption. 6.3 Nash Equilibrium Bidding Strategies Consider f i r s t the case of second-price auct ions . Since i n th i s r o - t a t i o n c r e d i t c o l l u s i o n model, a bidder i s purely a s e l l e r (unl ike the Hui bidder who i s both a s e l l e r and a buyer) , i t i s c l ear that the demand- revea l ing property of second-price auctions appl ies here. (See Vickrey (1961) and Cox, Roberson and Smith (1982).) Hence, Theorem 6 • 1 (Second-Price) : Under Assumptions 0 - 1 and C2 - C5, the s e c o n d - p r i c e Nash e q u i l i b r i u m bidding program b. for s e l l e r i i s given i n n by: b i n v. i n + G. 'in for n = 1, N - l , (6.3) and b i N = V i N - e < b w (6 .3 ' ) - 195 - where for n = 1, N-2, and G i , N - l = °« < 6 ' 4 , ) Proof: Omitted. For the case of f i r s t - p r i c e auct ions , we pos i t the fo l lowing recur - s i v e l y defined bidding program: b J = v, + G. + ( l /m)(v - v , ) , for n - l , N - l , (6.5) i n i n i n n i n and b i N = V i N = b W - ( 6 - V ) The d i s t r i b u t i o n of b . , F . , i s given by: i n i n 1-F. (b. ) = (v - v , ) / (v -v ) = n(v - b , +G, ) , i n v i n n i n n - n n i n i n where r\ = m/[ (m-1) (v^-v^) ] . C l e a r l y , the p r o b a b i l i t y of winning P ^ n i s given by P. (b. ) = [1-F, (b, ) ] m _ 1 . (6.6) i n i n i n i n J That (6.5) i s a Nash e q u i l i b r i u m bidding strategy can be demonstrated by showing that i t maximizes the d i s c o u n t e d expected g a i n from p a r t i c i p a t i o n i n the c u r r e n t and f u t u r e b i d d i n g s , G^ n_^> § i v e n by expression ( 6 . 2 ) . Hence, Theorem 6.2 ( F i r s t - P r i c e ) : Under Assumptions 0 - 1 and C2 - C5, the f i r s t - p r i c e Nash e q u i l i b r i u m b i d d i n g program b^ for the s e l l e r i i s given r e c u r s i v e l y by expression (6.5) with G^ given by expression (6 .4 ) . Proof: Omitted. - 196 - Note that the expected gains from p a r t i c i p a t i o n i n the bidding cons- p i r a c y G ^ n are equal under f i r s t - p r i c e or second-price auct ions . Consequ- e n t l y , the per period expected costs from the buyer's point of view under e i t h e r auct ion i n s t i t u t i o n are the same. This appears to correspond with V i c k r e y ' s r e su l t s for s i n g l e - o b j e c t auct ions . We do not expect th i s to remain the case i f we extend the model to incorporate r i s k avers ion i n l i g h t of the r e s u l t s of Cox, Roberson and Smith (1982)'s extension of V i c k r e y ' s model. - 197 - 7 CONCLUSION This i s a f i r s t attempt from the r a t i o n a l choice perspect ive at a r i - gorous understanding of the ' r o t a t i o n a l ' competit ive bidding process as a mechanism for in t ere s t rate formation, as wel l as loans and savings a l l o - c a t i o n s . To f a m i l a r i z e the reader with the Hui s t r u c t u r e , we began by g i v i n g several ac tua l examples and using small Hui to i l l u s t r a t e the i n - terworking of i t s s t r u c t u r a l parameters. We then introduced a d e f i n i t i o n of an agent's re servat ion discount vec tor , and demonstrated i t s existence and uniqueness under f a i r l y general circumstances. I t turned out to be p a r t i c u l a r l y useful for the d e r i v a t i o n of an optimal i n d i v i d u a l bidding s t ra tegy . In order to acquire t r a c t a b l e r e s u l t , we gave up a l o t of gene- r a l i t y to obtain the Nash e q u i l i b r i u m bidding program and the associated ex post borrowing and lending in t ere s t rates as wel l as the ex ante win- n ing-probabi l i ty -we ighted i n t e r e s t r a t e s . A side-reward from the Nash exercise Is an almost e x p l i c i t expression of members' surpluses for j o i n i n g the bidding market and, hence, of the t o t a l members' surplus , which provides a natura l candidate for an e f f i c i - ency measure of d i f f e r e n t r o t a t i n g c r e d i t markets. For example, loan a l l o c a t i o n s may be determined by l o t , by s e n i o r i t y , or by other ' s o c i o - l o g i c a l ' c r i t e r i a ( L i t t l e , 1957), not forge t t ing the neighborhood loan shark or pawn shop, the c r e d i t union, and the insurance companies .* 3 The We may also examine a d i s c r i m i n a t i v e vers ion of the r o t a t i o n a l bidding p r o c e s s where the winner of an a u c t i o n at p e r i o d n c o l l e c t s A - b j n rather than A - b n from bidder j . - 198- t o t a l members' surplus also provides the natura l objec t ive funct ion for the dual problem — the organ izer ' s problem — of the optimal combination of the membership. These are promising topics for immediate fol low-up research . Another d i r e c t i o n for future work concerns the r i s k sharing aspect of the r o t a t i o n a l bidding process . We can introduce an insurance dimension to the problem by adding an exogenous p r o b a b i l i t y of a large loss i n i n - come at any one p e r i o d , and study how the optimal bidding strategy may be modified to r e f l e c t the need to cover unexpected losses over time. Another refinement of the model i s to introduce r i s k avers ion into the Nash model ( e . g . , agents may have t ime-add i t i ve , constant r e l a t i v e r i s k aversion von Neumann-Morgenstern u t i l i t y funct ion with t h e i r r i s k avers ion indexes drawn from some known p r o b a b i l i t y d i s t r i b u t i o n ) . What about in troduc ing the p o s s i b i l i t y of default? T h i s , i f proper ly done, would considerably enr ich the current model and should provide a t h e o r e t i c a l explanation behind the t y p i c a l l y observed r i s k sharing s t r u c - ture of a r o t a t i n g c r e d i t a s soc ia t ion where the organizer bears the de- f a u l t r i s k of each member, but poses a common r i s k to the membership c o l - l e c t i v e l y . There appears to be a curious p a r a l l e l between agency theory and the r o t a t i o n a l bidding problem. The p r i n c i p a l ' s problem i s to e x p l i - c i t l y manipulate the agent's payoff s tructure r e l a t i v e to the informat ion- monitoring technology a v a i l a b l e , while the Hui organizer i m p l i c i t l y mani- pulates the member's payoff and his share of defaul t r i s k by opt imiz ing over the time preference, r i s k a t t i t u d e , and r i s k i n e s s of the other mem- bers over a v a i l a b l e membership poo l . The organizer can also attempt to change the rules such as bearing only ha l f of the default r i s k ins tead . - 199 - In the opposite d i r e c t i o n , a r o t a t i n g c r e d i t a s soc ia t ion in Vancouver, with a monthly loan pool of approximately $10,000, was formed j o i n t l y by i t s 20 members who are mutual f r i e n d s , completely doing away with the organizer (Chang, 1981). Beyond the immediate horizon i s the question of the ro l e of the i n - formal f i n a n c i a l sector whose c a p i t a l i s f inanced mainly by numerous r o t a - t i n g c r e d i t assoc iat ions with organizers ac t ing as information and r i s k a r b i t r a g e r s between the formal and the informal sec tors . This seems to be a reasonable approximation to the c a p i t a l market of Taiwan, Hong Kong, Singapore, and other east Asian countries and many A f r i c a n nat ions . One estimate puts the s i ze of Ethop ia ' s informal f i n a n c i a l sector at 8% of i t s GNP ( M i r a c l e , M i r a c l e and Cohen, 1980). Such a d e s c r i p t i o n may even apply , i n the developed countr i e s , to c e r t a i n pocket of the pupolat ion wi th in the o v e r a l l economy. In Hawaii today, the r o t a t i n g c r e d i t a s s o c i - a t i o n s , c a l l e d 'ko' among the Japanese, are s u f f i c i e n t l y prevalent as to be declared i l l e g a l . 1 4 Since A k e r l o f ' s "the market for lemons" paper (1970), we have under- stood a l o t more about the cause of market f a i l u r e s due to informat ional asymmetry which makes the formal f i n a n c i a l sector inherent ly imperfect . A not unreasonable conjecture may be that the informal sector complements the formal sector i n f inancing smal ler , shorter-term c a p i t a l , by Its r e l a - t i v e l y greater e f f i c i e n c y i n informat ion, monitoring, and even enforcement (which may, at times, be rather unorthodox). This seems to f ind at l east This information came from conversations with a Japanese American student from Hawai i . - 200 - s u p e r f i c i a l support i n the coexistence of the sophis t i ca ted f i n a n c i a l i n s - f i n a n c i a l i n s t i t u t i o n s of the formal sector i n the West along with the r o - t a t i n g c r e d i t a s soc ia t ion type f i n a n c i a l markets (among black West Indian immigrants in Brooklyn, New York; 'the very poor' i n San Diego; and i n centers of Asian immigration i n the developed countries ( M i r a c l e , Mirac l e and Cohen, 1980)) and other forms of informal f i n a n c i a l i n s t i t u t i o n s , such as the above-mentioned i n t r a - f a m i l y loans , loan sharks and pawn shops. In Sect ion 6, we described a simple way i n which c o l l u s i o n among a smal l group of s e l l e r s (s ince the gain from the r o t a t i n g c r e d i t c o l l u s i o n i s inverse ly re la ted to the number of p a r t i c i p a n t s ) i n a repeated auctions s e t t i n g can take place with minimal moni tor ing . This model may be modi- f i e d or extended i n a few ways. Instead of the pr ivate but known and de- t e r m i n i s t i c cost streams, we may assume that every s e l l e r draws from a d i s t r i b u t i o n his cost p r i o r to each bidding period and inves t igate the e f fec t s of various d i s t r i b u t i o n a l and informat ional assumptions about the costs on the r e s u l t i n g bidding s t r a t e g i e s . We may also assume that s e l - l e r s have add i t ive intertemporal u t i l i t i e s which are not neces sar i ly l i - near to study the e f fec t s of intertemporal r i s k avers ion . A d i f f e r e n t question re la ted to the c o l l u s i o n model i s whether there i s an incent ive to defaul t on the agreement once a s e l l e r has won i n a p a r t i c u l a r bidding p e r i o d . In Chew and Reynolds (1984), i t i s shown that r o t a t i n g c r e d i t c o l l u s i o n i s consistent with noncooperative Nash behavior i n an i n f i n i t e horizon framework under a symmetric uncerta in v a r i a b l e costs assumption. The l e n g t h - o f - c o l l u s i o n analys i s of Radner (1980) and the co l lus ion-wi th-bonding analys i s of Eswaran and Lewis (1983) may also be adapted to shed l i g h t on t h i s quest ion . - 201 - REFERENCES A k e r l o f G. A . , "The Market for 'Lemmons': Qua l i ty Uncertainty and the Mar- ket Mechanism," Quarter ly Journa l of Economics 84 (August 1970): 488- 500. Ardener, S. 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Reynolds, "A Rotat ing Credi t Structure for T a c i t C o l - l u s i o n i n Repeated Auctions with I n f i n i t e Hor izon ," U n i v e r s i t y of Arizona Working Paper, 1983. Chua, B. H . , "The Chit Fund in Singapore: F a i l u r e of A Large-Scale Rota- t i n g Credi t A s s o c i a t i o n , " i n Southeast A s i a : Women, Changing S o c i a l S tructure and C u l t u r a l C o n t i n u i t y , edited by G. B. Hainsworth, U n i v e r s i t y of Ottawa Press , 1981, 119-134. Coppinger, V . , V. L . Smith, and J . T i t u s , "Incentives and Behavior i n E n g l i s h , Dutch, and Sealed-Bid Auct ions ," Economic Inquiry 18 (January 1980): 1-22. Cox, J . C , B. Roberson and V. L . Smith, "Theory and Behavior of Single Object Auct ions ," i n Research i n Experimental Economics V o l . 2 , ed. by V. L . Smith, Greenwich: JAI Press , 1982. Cox, J . C , V. L . Smith, and J . M. Walker, "Theory and Behavior of M u l t i - ple Unit Auct ions ," Discuss ion Paper, Department of Economics, U n i v e r s i t y of A r i z o n a , 1982. Delancey, M. W., "Ins t i tu t ions for the Accumulation and R e d i s t r i b u t i o n of Savings among Migrants ," Journal of Developing Areas 12 (January 1978): 209-224. - 202 - Embree, J . F . , Suye Mura: A Japanese V i l l a g e , Chicage, 1939. Engelbrecht-Wiggans, R . , "Auctions and Bidding Models: A Survey," Manage- ment Science 26 (February 1980): 119-142. Eswaran, M. and T. R. Lewis, "Col lus ive Behaviour i n Repeated F i n i t e Games with Bonding," Cal tech S o c i a l Science Working Paper, 1983. Forsythe , R. and R, M. Isaac, "Demand Revealing Mechanisms for Pr ivate Good Auct ions ," Discuss ion Paper, Department of Economics, U n i v e r s i t y of A r i z o n a , September 1980. Geer tz , C , "The Rotating Cred i t A s s o c i a t i o n : A 'Middle Rung' i n Develop- ment," Economic Development and C u l t u r a l Change 10 ( A p r i l 1962): 241- 263. Green, J . and J - J L a f f o n t , "Character iza t ion of S a t i s f a c t o r y Mechanisms for the Reve lat ion of Preferences for Publ ic Goods," Econometrica 45 (March 1977): 427-438. H a r r i s , M. and A. Raviv , " A l l o c a t i o n Mechanisms and the Design of Auc- t i ons ," Econometrica 49 (November 1981): 1477-1499. Huang, Y. M . , personal correspondence, 1981. L i g h t , I . H . , Ethnic E n t e r p r i s e i n America, U n i v e r s i t y of C a l i f o r n i a Press , 1972. L i t t l e , K . , "The Role of Voluntary Assoc iat ions i n West A f r i c a n Urbaniza- t i o n , " American Anthropologis t 59 (1957): 579-596. Milgrom, P. R . , "A Convergence Theorem for Competitive Bidding with D i f - f e r e n t i a l Information," Econometrica 47 (May 1979): 679-688. Milgrom, P. R. and R. J . Weber, "A Theory of Auctions and Competitive B i d - d ing ," Econometrica 50 (September 1982): 1089-1122. M i r a c l e , M. P . , D. S. Mirac le and L . Cohen, "Informal Savings M o b i l i z a t i o n i n A f r i c a , " Economic Development and C u l t u r a l Change 28, (1980): 701- 724. Myerson, R . , "Optimal Auct ion Design," Mathematics of Operations Research 6 (1981): 58-73. Oren, M. E . and A. C. Wi l l i ams , "On Competitive B i d d i n g , " Operations Research 23 (November-December 1975): 1072-1079. Oren, S. S. and M. H. Rothkopf, "Optimal Bidding i n Sequential Auct ions ," Operations Research 23 (November-December 1975): 1080-1090. - 203 - Radner, R . , "Col lus ive Behaviour i n Noncooperative E p s i l o n - E q u i l i b r i u m of O l i g o p o l i e s with Long but F i n i t e L i v e s , " Journal of Economic Theory 22 (1980): 136-154. R i l e y , J . and W. Samuelson, "Optimal Auct ions ," American Economic Review 71 (June 1981): 381-392. R o t h s c h i l d , M. and J . S t i g l i t z , "Equi l ibr ium i n Competitive Insurance Markets: An Essay on the Economics of Imperfect Information," Quarter ly J o u r n a l of Economics (1976): 629-649. S tark , R. M. and M. H. Rothkopf, "Competitive Bidding: A Comprehensive B i b l i o g r a p h y , " Operations Research 27 ( M a r c h - A p r i l 1979): 364-390. V e l e z - I b a n e z , C. G . , Bonds of Mutual T r u s t : The C u l t u r a l Systems of Rota- t i n g C r e d i t Assoc ia t ions among Urban Mexicans and Chicanos, Rutgers, The State U n i v e r s i t y of New Jersey , 1983. V i c k r e y , W., "Counterspeculat ion, Auctions and Competitive Sealed Ten- ders ," Journa l of Finance 16 (May 1961): 8-37. Wi l son , R . , "A Bidding Model of Perfect Competit ion," Review of Economic Studies (October 1977): 511-518. , "Competitive Exchange," Econometrica 46 (May 1978): 577-585. , "Auctions of Shares," Quarter ly Journa l of Economics (November 1979): 675-689. - 204 - APPENDIX Proof of Lemma 3 .1 : By the d e f i n i t i o n of v^, V ' l . N - l * - U i< Y iN> - V where Y i , N - l = ^ i O ^ ' ^ r ^ ^ i l ^ - ' ^ ^ . N - l + ^ i . N - l ' ^ N - ^ ' and By m o n o t o n i c i t y of U. , U . ( Y . „ , ) decreases i n v. „ ., and U. (Y. . T ) i n c r e a -J J i ' i i , N - l i , N - l I iN ses i n Given monotonicity and p o s i t i v e time preference, we have V ^ V ^ V - i , N - l i , N - l i . N - 1 ' where v ^ ^ = - | I ^ - I ^ ^ | and v ^ ^ = (N+1)A. Let * = V^.N- lK.N- l^ i .N- l* ' 3 = V^ . N- lk . N - l ^ i . N - l * ' *• " U l < Y i H l v i f H - l " 2 ± , N - l ) » 5 = V^NKN-I^.N-I^ Again , p o s i t i v e time preference and monotonicity imply that a > d, ( A . l ) a > a, (A.2) d < d, (A.3) and a < d. (A.4) S i n c e i s assumed continuous, i n e q u a l i t i e s ( A . l ) , ( A . 2 ) , (A.3) and (A.4) imply that v^ ^ ^ ex i s t s and i s unique. By s i m i l a r arguments, i t can be es tabl i shed that , in general , there - 205 - exis ts a unique v . = ( v , v , v . „ , ) with v . e [v. , v . ] , where i i l 12 ' i , N - l i n - i n ' i n J v . = min{ l , . , - 1 , , T?—— [ I . - I , , .+ (N-n- l )v . , . ] } , - i n 1 i , n + l i n N-n 1 i n i , n + l i , n + l J " N+1 . A, i n N-n+1 for a l l n £ { l , . . . , N - l } , such that equation (3.1) holds . Q . E . D . Proof of Theorem 4.1: At p e r i o d n , b^ , k = 1 n - l , are known. Given t ime-addi t ive u t i l i t y , the c o n d i t i o n a l re serva t ion u t i l i t y of member i at period n i s K ^ € i x i k u i < i i k - ( A - v i k ) ) + \ J u i ( i i J + N A - ( N - j ) v i J ) + Cj+ixikui(Iik-A)' j = n, n+1, N. (A.5) Under assumptions 0, 1, 2, 3' and 4, the problem of member i at period n i s choos ing b. to maximize Euf ( b . ) g iven i n expression (4 .3 ) . Note i n i n i n that maximizing E U ^ n ( b £ n ) i s equivalent to maximizing E i n < b i n > = "XT" rEUin<binKl i n = J - E u f (b, ) A., i n i n i n " T~~ ^ i n U i ( I i n + N A - ( N - n ) V i n > + E k ! n + l X i k U i ( I i k - A ) H F i n ( b l n ) ] N - n in " TT~ ^ i n - i ^ i n - ^ - i n ^ K ^ H l - t F ^ C b ^ ) ] ^ } , in = {u i ( I . n +NA-(N-n)b i n ) -u i ( I i n +NA-(N-n)v l n ) } [F i n (b i n ) ] N - n + C ^ i ^ i n - ^ - i ^ i n - ^ i n ) + ^ + 1 ^ I n ^ * > in in (A.6) where - 206 - E ? j.i = E 4 .i_i(b* = Max E . ..(b. . . ) • i , n + l i , n + l i , n + l i , n + l i , n + l The f i r s t order condi t ion for b^ n to maximize expression ( A . 6 ) , a f ter suppressing the i n d i v i d u a l and time s u b s c r i p t s , i s 0 = E ' (b ) = - (N-n)F(b) N " n ~ 1 {u , ( I+NA-(N-n)b)F(b) - ( | )F , (b) }, (A.7) where <|> = u(I+NA-(N-n)b) - u(I+NA-(N-n)v) - [u(I-A+b)-u(I-A+v)+E*] , (A.8) and E * = ( j . i / K )E* J.I • i , n + l i n i , n + l Note that (A.7) implies equation ( 4 . 5 ) . With increas ing u t i l i t y , the RHS of equation (4.5) i s increas ing in b. With decreasing marginal -outbidden-rate d i s t r i b u t i o n F ( b ) , the LHS of equation (4.5) i s decreasing i n b. Therefore , the bid that s a t i s f i e s (4.5) i s unique. The second order condi t ion for b to maximize (A.6) requires that (N-n)u"(I+NA-(N-n)b)F(b) - [(N-n+l)u*(I+NA-(N-n)b)+u'(I-A+b)]F'(b) + <t>F"(b) < 0, which i s s a t i s f i e d l o c a l l y by the b that solves equation (4.5) i f F y i e l d s decreasing marginal outbidden r a t e . Hence, the b id that maximizes (A.6) i s unique. Q . E . D . - 207 - Proof of Corollary 4.5; ob* We want to show t h a t the s i g n of i s p o s i t i v e . Take t o t a l d e r i v a t i v e of both sides of equation (A.7) to obtain 0 = E"(b*)(ob*) + 5 E ' ( b * ) (ov) ov = E"(b*)(ab*) + ( N - n ) [ F ( b * ) ] N ~ n ~ 1 F ' ( b ) ( | i ) ( 5 v ) . Therefore | £ = - { ( N - ^ t F C b ^ l ^ - S ' C b ^ ^ l / E ^ C b * ) , where = (N-n)u'(I+NA-(N-n)v) + u'(I-A+v) > 0. ov 9b* Since E"(b*) < 0 and F ' (b ) > 0, we have > 0 as d e s i r e d . The proof for the second-price case i s s i m i l a r . Proof of Corollary 4.7; Take the t o t a l d e r i v a t i v e of (A.7) to obta in dE! (b* ) 0 = E" (b* )(3b* ) + [— X" — ] ( 9 X , ) i n i n / v i n L oX J i n i n which impl ies 9b* 9E*. (b* ) . i n m v i n ' 1 ax. oX. E" (b* )' i n i n i n i n ' Since 9E'. (b* ) i n i n Q . E . D . = (N-n)[F, (b* ) ] N n h\ (b* ) - E * . • T4^ 2 > °» ^ L i n v m J i n i n ' i ,n+ l (X ) * ax, ^ _ . . . . . . . , i n ' i n y and E'^ n (b* n ) < 0, we have shown that a D * n / a ^ n > 0 < The proof of the second-price case i s s i m i l a r . Q . E . D . - 208 - Proof of Theorem 5 » 1 ; In this proof, without fear of confusion, we w i l l omit the time index n to simplify the notation. Suppose, at period n, bidder j believes that a l l his r i v a l s adopt the b i d d i n g strategy function (5.3). By assumption 4', v^, for a l l i * j , i s uniformly d i s t r i b u t e d over [v,v]. Since b^ i s l i n e a r i n v^, by Assumption 4', b^ w i l l be uniformly d i s t r i b u t e d over [b,b], where b = v - -i- G*, — - m i b - v - - ^ r - (v-v) - - G*. m+1 - m i R e c a l l that m = N-n+1 i s the number of bidders at period n and that GJ i s the i d e n t i c a l s u b j e c t i v e l y discounted expected gain from future bidding p a r t i c i p a t i o n . The inverse of (5.3) i s v, = f (b, ) = — [b - -L_ v + - G*] . i i m • i m+1 - m x Therefore, f ( b ) v-v F,(b.) = F (v.) = / d[ ] = n [b, — v + — G*], b l v i Jv L - i - m i - v-v and F^(b ±) = n = (^t!)/(v-v). Under Assumptions 0, 1, 2, 3", 4' and 5, the expected u t i l i t y d i f f e r - ence for bidder j of a bid b. i s E.(b.) 3 J = (m-l)(v -b )[n(b -v+ I G * ) ] m _ 1 + (m-l)j£ (x-v +G*)[n(x-y+ I G*]m"2ndx, J J J J j J J J - 209 - = ( m - l ) T i m 1 { ( v . - b . ) ( b -v+ - G * ) m 1 + ( x - v . + G * ) ( x - v + I G*)m~2dxV. (A.9) 1 j J J - m 2 J J - m j The f i r s t order condi t ion for b*. to maximize E . i s J J 0 = E'.(b*) 3 J = ( m - l ) T i m " 1 ( b * - v + I G * ) m ~ 2 ' [ - ( b -v+ - G*)+(m-l)(v .-b . ) - (b -v .+G*) ] 2 ~ m 3 J ~ m J J J J J J = (m-l)r | m " 1 (b*-v+ i G*)m" 2[-(m+l)b*+mv.+v- — G*] , (A.10) which impl ies b* = (mv. + v - — G*) = v . - - ^ p (v . -v ) - - G * . ( A . l l ) 2 m+1 j - m 2 J m+1 J " m j It i s s tra ightforward to check that E"(b*) = - ( m - l ) ( m + l )T i m " 1 [ b * - v + - G * ] m ~ 2 < 0 3 3 3 ~ m J s i n c e m > 1. Thus b* a l s o s a t i s f i e s the second order c o n d i t i o n . Since J bidder j ' s optimal s trategy ( A . l l ) i s the same as a l l h is r i v a l s ' , we have shown that the strategy funct ion (5.3) i s indeed a Nash equal ibrium bidding strategy at period n . I t then f o l l o w s that the vector b. = ( b . . , b . _ , . . . , b . , ) , with b. i i l ' i 2 ' ' i , N - l ' i n g iven by (5 .3 ) , where n = 1, 2, N-1, i s a Nash e q u i l i b r i u m bidding program. Q . E . D . - 210 -
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