TWO , ESSAYS ON FINANCIAL ECONOMICS: WEIGHTED UTILITY, RISK AVERSION AND PORTFOLIO II. COMPETITIVE BIDDING AND CHOICE INTEREST RATE FORMATION IN AN INFORMAL FINANCIAL MARKET by MAO, MEI HUI JENNIFER B.Comm., The N a t i o n a l Taiwan U n i v e r s i t y , 1975 M.B.A., The N a t i o n a l Cheng-Chi U n i 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 ( F a c u l t y of Commerce and B u s i n e s s A d m i n i s t r a t i o n ) We accept t h i s t h e s i s as conforming to t h e r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June 1985 © Mao, Mei H u i J e n n i f e r In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of requirements f o r an advanced degree a t the the University o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it f r e e l y a v a i l a b l e f o r reference and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f department o r by h i s o r her r e p r e s e n t a t i v e s . my It i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my permission. Department o f Commerce & Business Administra t i o n The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date /«i ^ July 16, 1985 written S u p e r v i s o r : D r . Robert A . Jones Abstract This thesis consists problem involving some aspects of Essay deals the general question I risk are with aversion robust with generalizing axiomatic weaker, and mean respect expected requirement consistent of and financial whether classical based on expected recent works in We called extended the the utility a research economics. results utility nonlinear investigate weighted equivalence i.e., spread, is and the also risky concave about utility theories implications theory in hypothesis along asset to of with an the risky-asset preference ordering stochastic individual be first equivalent, demand, without functional. complete, dominance, between the necessary a normal lottery-specific counterpart, of and any The only transitive, continuous two definitions is sufficient condition for in also context. the be definitions conditional certainty the preference The e q u i v a l e n c e identify of three conditional first-degree to a comparative among i n terms assumptions that with distribution. utility uncertainty choice utility. aversion, differentiability with Each essay addresses but unaxiomatized l i n e a r Gateaux u t i l i t y . preserving single to generalization risk We two e s s a y s . portfolio We e s t a b l i s h global of good utility which depends and to a weighted functions. only on the ii utility Unlike agent's its initial the maximizer expected wealth and preferences, risky this also depends on the characteristics of ding second process in corresponding informal essay examines the endogenous allocation financial of the of a sequential determination loans markets role and savings called the of competitive interest rates i n a widely 'rotating observed credit and t i m e - a d d i t i v e After the tions, deriving individual including equilibrium borrowing, the winning as risk well each bidding neutrality, strategy. as winning auction. in optimal the class association'. concave functions. some comparative bidding realized won the utility bid- and O p t i m a l b i d d i n g s t r a t e g i e s are o b t a i n e d f o r i n d i v i d u a l agents w i t h of the asset. The of condition bids, strategy, to This lending, statics obtain yields, interest including the rate one and e f f i c i e n c y we a impose further tractable for each depending for the implications restric- form of agent, on the period a Nash an ex post history of i n which he Weighted by the Nash e q u i l i b r i u m - i n d u c e d p r o b a b i l i t y of period, ex ante borrowing result. H i and lending interest rates TABLE OF CONTENTS ABSTRACT ii ESSAY I WEIGHTED UTILITY, RISK AVERSION AND PORTFOLIO CHOICE 0 INTRODUCTION 0.1 0.2 0.3 1 1.2 1.3 1.4 2 2 Expected U t i l i t y and F i n a n c e : H i s t o r y A l t e r n a t i v e Preference Theories 6 O r g a n i z a t i o n of the Essay 10 INDIVIDUAL RISK AVERSION 2.1 Local Risk Aversion 2.2 G l o b a l Risk A v e r s i o n 4 COMPARATIVE RISK AVERSION 40 58 67 4.1 D e f i n i t i o n s 67 4.2 C h a r a c t e r i z a t i o n s 72 DECREASING RISK AVERSION AND THE NORMALITY OF RISKY-ASSET DEMAND WITH DETERMINISTIC WEALTH 87 5.1 5.2 7 33 34 PORTFOLIO CHOICE PROBLEM 6 14 L i n e a r Gateaux U t i l i t y , L i n e a r I m p l i c i t U t i l i t y and Weighted Utility 15 Frechet D i f f e r e n t i a b l e U t i l i t y 18 Representation 20 S t o c h a s t i c Dominance 27 3 5 2 PREFERENCE REPRESENTATION AND STOCHASTIC DOMINANCE 1.1 1 Expected U t i l i t y 87 Non-Expected U t i l i t y 89 COMPARATIVE AND DECREASING RISK AVERSION INVOLVING STOCHASTIC WEALTH 103 6.1 Expected U t i l i t y 6.2 Beyond Expected U t i l i t y CONCLUSION 117 REFERENCES 124 105 110 iv ESSAY II COMPETITIVE BIDDING AND INTEREST RATE FORMATION IN AN INFORMAL FINANCIAL MARKET 130 0 INTRODUCTION 1 THE GENERAL STRUCTURE AND ACTUAL CASES OF HUI 1.1 1.2 2 131 The General S t r u c t u r e A c t u a l Cases of Hui of Hui 141 138 138 THE ECONOMICS OF HUI WITH TWO OR THREE MEMBERS 157 Two-Member Hui without an O r g a n i z e r 157 Two-Member Hui with an O r g a n i z e r 161 Three-Member Hui without an O r g a n i z e r 162 3 THE MODEL FOR AN N-MEMBER HUI WITH AN ORGANIZER Notations Assumptions 4 166 166 OPTIMAL INDIVIDUAL BIDDING STRATEGIES 4.1 4.2 4.3 166 F i r s t - P r i c e Competitive Bidding Second-Price Competitive Bidding Implications 177 171 172 175 5 A NASH PROCESS OF INTEREST RATE FORMATION 6 AN APPLICATION TO COLLUSION AMONG SEVERAL SELLERS UNDER REPEATED AUCTIONS 192 6.1 6.2 6.3 7 183 The S t r u c t u r e of R o t a t i n g C r e d i t C o l l u s i o n Assumptions 193 Nash E q u i l i b r i u m B i d d i n g S t r a t e g i e s 195 CONCLUSION 198 REFERENCES 202 APPENDIX Proof of Proof of Proof of Proof of Proof of 205 Lemma 3.1 205 Theorem 4.1 206 C o r o l l a r y 4.5 208 C o r o l l a r y 4.7 208 Theorem 5.1 209 v 192 LIST OF TABLES Table 0.1: Examples of R o t a t i o n a l T a b l e 1.1: Cash Flow P a t t e r n s of Table 1.2: A c t u a l Cases of Table 2.1: Ex Post I n t e r e s t Rates and P r o f i t s Example 160 Table 3.1: Participant i's Indifferent D i s c o u n t - B i d Hui 168 Table 5.1: Nash B i d d i n g S t r a t e g i e s and T h e i r D e r i v a t i v e s Hui Exchange 133 Hui P a r t i c i p a n t s 140 143 vi for A 2-member Hui - An Cash Flow P a t t e r n s i n a 188 ACKNOWLEDGMENTS I wish to thank my s u p e r v i s o r y committee - P r o f e s s o r s Robert Jones, Neal Stoughton, and John Weymark - as w e l l as P r o f e s s o r s A . Atnershi, Mukesh Eswaran and Hugh Neary for t h e i r h e l p f u l comments and s u g g e s t i o n s . I am e s p e c i a l l y g r a t e f u l to my s u p e r v i s o r D r . Robert Jones and P r o f e s s o r John Weymark f o r t h e i r support and encouragement. I have a l s o b e n e f i t e d from v a l u a b l e d i s c u s s i o n s w i t h D r . Chew Soo Hong. The s e c r e t a r i a l a s s i s t a n c e of Miss C o l l e e n C o l c l o u g h i s deeply a p p r e c i a t e d . ESSAY WEIGHTED U T I L I T Y , RISK I AVERSION AND PORTFOLIO CHOICE 0 INTRODUCTION 0.1 Expected U t i l i t y and F i n a n c e : H i s t o r y Given needs the tractable, certainty. sions, for of behavioral ria the topics rich it studies, enough t h e o r i e s simplicity, expected making under etc., however In response, and the expected finance about value two doubts theory, discipline preferences under u n - once popular as insurance, on i t s approaches, utility as Investors' types of cast a was uncertainty. i n the purchase of v a r i o u s portfolios, analysis the aver- diversi- theoretical namely emerged risk as the a and mean-vari- improved crite- for d e c i s i o n making under u n c e r t a i n t y . impact investigated on Tobin finance (1958), theory. (1961) linear asset and derived its market has von works laid the assumptions, the capital between risk. l e d to utility Neumann (1789), the its which and mean-variance of Sharpe expected widespread (1964), rate and Morgenstern - 2 - of Lintner in and by its and portfolio (1965) and establishes r e t u r n on analysis axiomatized (1947), 1959) modern which had intuitiveness acceptance to mean-variance first for p r i c i n g model simplicity was analysis Markowitz (1952a, foundation asset the The and complementary expected by since relationship Parallel revived has after certain variance analysis of by Tetens only and Under Treynor ment its validity. First the to decision evident fication of and yet Due criterion ance nature a risky of mean- finance. is the Ramsey refined by develop(1926), Marshak (1950), Samuelson Anscombe (1970), and pected Aumann Fishburn plausible axioms utility certainty its that last three with for Schlaifer (1964), DeGroot D e r i v e d from a set representation the of form, microeconomics another can always i n the dominated (1969) suffers of exun- that sense i n the dominance always that, delivers given better-than by a l o t t e r y from s e v e r a l and F e l d s t e i n stochastic lottery construct analysis Borch violates probability) stochastically (1954), tractable foundation analysis outcome a higher Savage and o t h e r s . Specifically, may be p r e f e r r e d to is and (1971) (1953), decades. lottery which Raiffa t r a c t a b i l i t y , mean-variance mean-variance one Pratt, p r o v i d e d the showed curve, Milnor lead to a s i m p l e , weaknesses. ence and Arrow theoretical that Herstein (1963), (1970), has i n the Despite (1952), on the (1969) (i.e., a a better an i n d i f f e r - region a given lottery indifference curve. It is variance holds: tions; also well analysis (1) is agents (2) known valid have that, only under if quadratic one von expected of the utility following theory, two mean- assumptions Neumann-Morgenstern u t i l i t y the u n d e r l y i n g random v a r i a b l e i s n o r m a l l y d i s t r i b u t e d . funcEither Borch (1969) showed t h a t , g i v e n two l o t t e r i e s A and B i n d i f f e r e n t i n the mean-variance sense, we can always c o n s t r u c t another p a i r of l o t t e r i e s A ' and B such that u.^ = |J^i , o" = a^, , Mg = Mgi , Og = Og , but A' s t o c h a s t i c a l l y d o m i n a t e s B' . S p e c i f i c a l l y , l e t u.^ > Lig, o^ > Og, and p = 1 A f Xa ( V ^ ^ A - ^ A - ° B > ^ A - ^ ^ V V ' > V H X B> ^1 ^ A °A^ > Y2 ^ B B/^1^ A' i l o t t e r y of g e t t i n g x with 1-p c h a n c e and g e t t i n g y^ w i t h p chance and B' i s the l o t t e r y of g e t t i n g x w i t h 1-p c h a n c e and g e t t i n g y2 w i t h p chance, then A' and B' are such ( ) = two 2 / [ + ( ) X 2 = + ( a ) 1 2 ] X + C J = x s lotteries. - 3 - t n e = = requirement overly is not satisfactory. restrictive function, on domain and with the expected risk risk Arrow-Pratt via the constant, or operations by increasing Arrow independently lute r i s k averison and terized, of i n terms risky-asset wealth is as Pratt the theory if (1964) and of premium and, risk further f o r an absolute/relative risk identified and re- known as the risk exhibit as aversion He also decreasing, well as the such p r o p e r t i e s . justifications for decreasing increasing r e l a t i v e r i s k aversion. decreasing investor's utility index. aversion, absolute risk aversion, demand i n a o n e - s a f e - a s s e t - o n e - r i s k y - a s s e t the i s now comparative which w i l l Pratt absolute the Arrow-Pratt functions (1971) expected f u n c t i o n u, former characterized Arrow aversion. the measure of r e s p e c t i v e l y (the established deterministic. utility aversion. theory. - x u " ( x ) / u ' ( x ) as preserve quadratic times that have risk c l a s s of u t i l i t y at f o l l o w i n g are l i n e s of r e s e a r c h risk Neumann-Morgenstern u t i l i t y He that w i l l The is a bounded known aversion, index). phenomena. r i s k premium, p r o b a b i l i t y premium, and identified good the aversion indexes - u " ( x ) / u ' ( x ) and local distribution i t requires utility now d e c i s i o n maker with von lative finance i n c r e a s i n g absolute literature characterized normal hand, i s unappealing because area of f i n a n c e , ties started other implies In the direct the in modelling The In addition, relative risk the safe the also charac- normality of world where i n i t i a l asset aversion He abso- is will be a increasing luxury in his wealth. Cass and Stiglitz (1970; wealth e f f e c t s i n t o the world the agent's p r e f e r e n c e 1972) extended of many r i s k y assets. e x h i b i t s a 'separation - 4 - Arrow's investigation of They showed t h a t , i f property' so that the risky assets as result will also a whole continue identified property sion. can be viewed to the either Hart to hold class be (1975) the Arrow a world for of agents' Mirman's natural tiple further a mutual these utility or that for m u l t i p l e and P r a t t ' s of generalization that have constant the the separation relative separation They risk property functions and thereby savings risk aver- is both r i s k y assets to be normal. characterization and collectively. to those representing K i h l s t r o m and Mirman (1974) were able many commodities consumption then Arrow's n o r m a l i t y assets functions display proved fund, risky by r e s t r i c t i n g u t i l i t y definition of investigate choice. aversion, comparative risk its to aversion implications M o t i v a t e d by K i h l s t r o m Paroush (1975) further of A r r o w - P r a t t ' s r i s k premium to ge- and proposed the world of a mul- commodities. In another Kihlstrom, vestible that, (in of same o r d i n a l p r e f e r e n c e s , neralize to for quadratic n e c e s s a r y and s u f f i c i e n t Alternatively, as i n i t i a l wealth the the sense counterpart. is case of of premium to Ross (1981) the i n i t i a l wealth While Arrow and P r a t t of the u t i l i t y Rothschild and (1978), the Ross (1981), case where against of i n i t i a l wealth, individual a given (1981) proposed even i f Kroll investigated deterministic insure characterization aversion, and random r a t h e r than d e t e r m i n i s t i c . Arrow-Pratt) P r a t t 's properties Levy Romer and W i l l i a m s unlike higher direction, need risk a stronger comparative risk not and the It was in- shown a more r i s k averse be willing than h i s less to risk pay a averse c o n d i t i o n under which Arrowaversion will c a r r y through i s random. focused their attention on the r e l a t i o n between f u n c t i o n and the b e h a v i o r a l i m p l i c a t i o n s Stiglitz (1970; - 5 - 1971) characterized a of risk decision maker's risk, risk a v e r s i o n by h i s termed utility response 'mean p e r s e r v i n g functions are e q u i v a l e n t Moreover, each of criterion i n the latter. Following cept, risk tain Rothschild They showed general than and Stiglitz' proposed useful While expected seems to have served by d e c i s i o n of its thin an i n t e r v a l of monetary outcomes. ence of these phenomena, the quiet nance should not be i n t e r p r e t e d illustrated viable below, alternative several it alternative spread called the con- 'mean comparative that of - 6 - light prevalence expected of cer- Allais been in and persist- utility in fi- R a t h e r , as w i l l be tractability We w i l l , have empiri- i n c l u d i n g the as unreserved c o n t e n t . theories in its and r i s k - a v e r t i n g b e h a v i o r w i - Given the theory. assum- utility. finance w e l l , implications, its consti- c r i t i c a l l y on the scientists acceptance i s m a i n l y due to preference i m p l i e d by, preserving expected Theories reported v i o l a t i o n s variance not mentioned here A l t e r n a t i v e Preference widely not for c h a r a c t e r i z i n g the agent i n q u e s t i o n maximizes h i s questioned popular a similar notion on r i s k a v e r s i o n which r e l i e s been once but i s mean along with others has of are l i n e a r i n p r o b a b i l i t y d i s t r i b u t i o n . paradox and the concurrence of r i s k - s e e k i n g upon the former i m p l i e s , The above validity concavity aversion. spread' which i s utility in i n 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 that literature the increase mean p r e - (1974) investigations that of for a d i s t r i b u t i o n over i t s more the a p a r t i c u l a r type aversion for preferences p t i o n that cal is sense that preserving t u t e the 0.2 these Diamond and S t i g l i t z utility spread'. and the p r e f e r e n c e s e r v i n g spreads to and the this reported l a c k of a section, i n the touch litera- ture and then, in the remainder of this essay, which h o l d promise to open some new paths nomics beyond expected account line of f o r the i.e. investigations, the form form approaches f i n a n c i a l eco- probabilities) adherents to by Edwards (1954), utility anomalies utility E p ^ u ( x ^ ) , by a n o n l i n e a r Subsequent for i n v e s t i g a t i n g started r e p o r t e d expected some s u b j e c t i v e two Theories b a b i l i t y weights p^ i n the expected ry, on utility. Misperception-of-Probability This focus which may not this view by r e p l a c i n g the expression function f(p^) for a f i n i t e Handa lotte- lotteries before deciding (1977), who and Kahneman and Tversky (1979), who which of their two as unity. adopted Ef(p^)x^, Karmarkar (1978), who n o r m a l i z e d the weights u s i n g E[ f (p^ ) / E f ( p^, ) ] u ( x ^ ) , to pro- (often interpreted add up to include attempts evaluation the edited equations to be applied. There tation nite are takes at the lotteries. tinuous over /f(p^,x^). 2 continuous their This quence three [a,b]. Since this, It is finance distributions, is this First suppose not all, it finite t r i b u t i o n does not l i m i t of lotteries can only apply to known how we can c a l c u l a t e l i m i t a t i o n that the v a l u e converging exist. 7 - Ef(p£,x^) to con- involving c a s t s doubt on rule. or / f ( p ^ , X £ ) a continuous fi- E f ( p ^ , x ^ ) or concerned with problems a serious - represen- the p r o b a b i l i t y d i s t r i b u t i o n i s frequently is of any approach whose f i n a n c i a l economists as a g e n e r a l d e c i s i o n because the of problems w i t h form £ f ( p ^ , x ^ ) . To see appeal to is least of any probability sedis- Another to violate problem with stochastic any Edwardsian theory dominance. detection-of-dominance its inherent Kahneman and Tversky operation steep p r i c e i n v i o l a t i n g is to circumvent t h i s transitivity (Chew, 1980; (1979) tendency added d i f f i c u l t y but Machina, the paid a 1982a). The t h i r d problem w i t h 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 in its inability to i n which case the sense of spreads under expected preference (i.e. for utility a v e r s i o n towards utility obtains. a distribution distributions expected aversion is d i s p l a y g l o b a l r i s k a v e r s i o n except when f(p^) with over equal equivalent to Global any mean but pointwise actuarially fair infinitesimal to r e p l a c e expected General Preference function cal paradoxes a preference He argued that of value' the mean-preserving- higher local in variability) risk risks). aversion is (i.e. Since g l o b a l financial risk econo- which are property. Functionals M o t i v a t e d by the utility. its aversion, u t i l i t y by t h e o r i e s i n h e r e n t l y unable to d i s p l a y such a p r e f e r e n c e consider of regarded as an a p p e a l i n g p r o p e r t y i n f i n a n c e , m i s t s might be r e l u c t a n t to risk = p^, i n his functional name, more A l l a i s is general than a d e c i s i o n maker's p r e f e r e n c e the moments of the functions. Suppose d i s t r i b u t i o n of V is the some Allais' probably the is that of first expected represented by a 'cardinal psychologi- preference functional. Then V = V(m^,m ,...), 2 where m. = 1 Js(x) dF(x). T h i s model is indeed v e r y g e n e r a l - 8 and c o n t a i n s - both expected utility and mean-variance only on the expected analysis first inability because h i g h e r moments tion analysis s(x) first is is VCm^jii^) — linear loping results in the interesting becomes a when the preference u t i l i t y obtains. describe Allais-type cardinal depends In A l l a i s ' choice Another s p e c i a l outcome view, behavior is case — mean- psychological u n d e r l y i n g monetary the success that to note preference and t e n a c i t y of the also v i o l a t e Two other pected value x, func- and o n l y the weighted analysis' utility without 1982; representation finance. asset pricing model V = mean-variance i n deve- in light capital i l l u s t r a t i o n , however, like then d i r e c t i o n is mean-variance-based functionals theory r e c e n t l y 1981; nonlinear, for a ' u t i l i t y m e a n - v a r i a n c e ' to expected the it of asset is clear analysis, will 1983), forms, utility also belong i n - one i s Machina's (1982a) independence axiom, the other the 'exis proposed i n Chew and MacCrimmon (1979a & of the essay w i l l be devoted sumptions, ing to present theories of g e n e r a l p r e f e r e n c e Chew (1980; The r e s t capital analysis, is dominance. alternative utility s(x) One p o s s i b l e Based on Borch (1969)'s stochastic if functional to be e x p l o r e d . utility-mean-variance category that, 'utility-mean-variance' p r i c i n g model. a to When the are unduly i g n o r e d . a n a l y s i s — a case yet b), cases. two moments m a t t e r . It the special moment m^ , expected utility's variance as Fishburn (1983) and Nakamura to d e t a i l e d d i s c u s s i o n s as w e l l as results (1984). on t h e i r potentially as- interest- J Other approaches not c o n s i d e r e d here i n c l u d e the i n f o r m a t i o n p r o c e s s i n g models ( e . g . Payne, 1973), M e g i n n i s s ' (1977) ' e n t r o p y ' p r e f e r e n c e , and the theory of r e g r e t proposed by B e l l (1982) and Loomes and Sugden (1983). 9 - 0.3 O r g a n i z a t i o n o f the E s s a y T h i s essay focuses on comparisons of three p r e f e r e n c e l y expected of their Section weighted conditions, their technicalities, consistency any u s e f u l with and l i n e a r Gateaux u t i l i t y , and implications representation they are a l l first-degree preference ty under d i f f e r e n t utility properties 1, we summarize mathematic the utility, stochastic are risk sion. The d i s t i n c t i o n expected aversion, utility. them are r i s k tainty aversion equivalent, pointwise risk equivalent shown to be result in regardless only r e q u i r e s We regard a property that proper- and i m p l i c a t i o n s of indi- a v e r s i o n and g l o b a l r i s k aver- they of aversion risk do not coincide beyond are d e f i n e d . Among c o n d i t i o n a l and u n c o n d i t i o n a l terms the of of tion. some of contrary, under expected 3 introduces safe asset and one will never short risky sell the the preference stochastic Section To avoid mean p r e s e r v i n g cer- spread, R i s k a v e r s i o n i n terms of c o n d i t i o n a l s i s t e n t with f i r s t - d e g r e e equivalent forms. In and cer- and r i s k a v e r s i o n i n terms of mean p r e s e r v i n g spread are equivalent On the terms aversion. hypotheses. because sense of aversion local risk aversion. tainty This notions i n the in The c o n d i t i o n for t h i s risk important Different as name- given. including local is risk dominance as In S e c t i o n 2 , we i n v e s t i g a t e the p r o p e r t i e s vidual of functional stated theory must p o s s e s s . theories theories, underlying preference to be complete, transitive, dominance and continuous the risk aversion theory. con- in distribu- definitions are only in with utility. portfolio asset. choice It is r i s k y asset - 10 problem showed as - long that as its a world a risk averse expected one agent return is strictly greater strategy than the and return. When i s to i n v e s t only i n the s a f e a s s e t . maximizer, then he w i l l if safe only they I f he are equal, his i s a weighted best utility i n v e s t a p o s i t i v e amount i n the s i n g l e r i s k y i f i t s expected return is strictly greater than asset the safe return. We if f u r t h e r show t h a t , i f an agent w i l l i t s expected return must be g l o b a l l y r i s k lent. averse greater i n the any r i s k a v e r s i o n and preference ordering than the only safe r e t u r n , then he sense of c o n d i t i o n a l c e r t a i n t y e q u i v a - T h i s amounts to the e q u i v a l e n c e ty-equivalent for is s t r i c t l y i n v e s t i n the r i s k y asset between g l o b a l c o n d i t i o n a l - c e r t a i n - global conditional portfolio satisfying completeness, risk aversion transitivity and f i r s t - d e g r e e s t o c h a s t i c dominance. S e c t i o n 4 c h a r a c t e r i z e s comparative r i s k a v e r s i o n across Again, we certainty prove that equivalent of mean p r e s e r v i n g comparative risk is equivalent spread without aversion in to comparative terms risk individuals. of conditional aversion depending on s p e c i f i c u t i l i t y i n terms functional forms. In the Section risky This r e s u l t i s not 5, asset to al ted a the normal the e x p l i c i t necessary good and and comparative risk l i n e a r Gateaux u t i l i t y results sufficient a weighted in this condition utility for maximizer. and utility. s t o c h a s t i c wealth i s i n t r o d u c e d r i s k s because t h e i r u t i l i t y utility for and f u n c t i o n a l form of weighted u t i l i t y , o b t a i n a b l e under l i n e a r Gateaux decreasing utility be utilizes In S e c t i o n 6, of we .derive aversion. It to the c h a r a c t e r i z a t i o n appears that weighted provide more room i n a l l o w i n g a d d i t i o n - functions s e c t i o n are are l o t t e r y - s p e c i f i c . from Ross (1981). The The expec- results be- yond expected u t i l i t y fferentiable utility. utility were f i r s t and l a t e r proved by Machina extended They are reproduced mainly (1982b) f o r Frechet d i - by Chew (1985) to l i n e a r to complete the spectrum Gateaux o f our com- parisons . In this essay, definitions, e x p r e s s i o n s , lemmas and c o r o l l a r i e s are numbered a c c o r d i n g to the s e c t i o n and the order i n which the convenience of comparisons rems are i n g e n e r a l l a b e l l e d number. O b v i o u s l y , EU, WU utility, linear w i t h U, EU, WU and LGU Gateaux u t i l i t y , does not depend on a s p e c i f i c ber i n d i c a t e s across d i f f e r e n t stand Theorem No. preference theories, For theo- or LGU, f o l l o w e d by an A r a b i c f o r expected respectively. utility, weighted U i s used when the r e s u l t preference functional the nature o f the r e s u l t . they appear. form. The A r a b i c num- A summary i s g i v e n below: Regarding 1 representation 2 first-degree 3 Arrow-Pratt 4 pointwise l o c a l r i s k aversion 5 global r i s k aversion 6 nonnegative or 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 7 comparative 8 d e c r e a s i n g r i s k a v e r s i o n w i t h d e t e r m i n i s t i c wealth 9 comparative s t o c h a s t i c dominance (SD) index (PLRA) (GRA) demand r i s k a v e r s i o n w i t h d e t e r m i n i s t i c wealth (CRA) r i s k aversion with s t o c h a s t i c (DRA) wealth 10 decreasing r i s k aversion with s t o c h a s t i c but d e t e r m i n i s t i c wealth increments initial wealth 11 d e c r e a s i n g r i s k a v e r s i o n with s t o c h a s t i c and wealth increments initial wealth - 12 - When there tions are are used Section more of one theorem on the i n l a b e l l i n g them. 7 which applications than then concludes non-expected same s u b j e c t , These r e s u l t s this essay utility theories economics. - by 13 - decimal frac- are b r i e f l y summarized i n suggesting in the some area of potential financial 1 P R E F E R E N C E To R E P R E S E N T A T I O N obtain normatively ence with sights which into the utility without D denote > V(G). interest. F the the identify then the very little some p r e f e r e n c e be is of useful set one a priori can res- successively expected in of a prefer- Alternatively, i m p l i c a t i o n s of latter a construct h i s t o r i c a l development space utility providing i n - i m p l i c a t i o n s which are d e r i v e d sensitive to the other 0 to p r o b a b i l i t y measures The weakest customary completeness ordering is underlying on some theore- requirement outcome for and t r a n s i t i v i t y . represented by a u t i l i t y This r u l e s out any l e x i c o - g r a p h i c type representation desirable = ( l - a ) F + OG, words, 1. F 'direction' F a t t is this level of generality s t r u c t u r e to impose on V , where a e from F to G. a F . The f i r s t - a preferWe suppose and o n l y i f preferences. is of little define [ 0 , 1 ] . a p r o b a b i l i t y mixture of F and G. goes ° of at set f u n c t i o n a l V: for any F , G e D, F i s weakly p r e f e r r e d to G i f To i d e n t i f y a of restrictions. the A preference from can axioms. ordering with approach, such a p r e f e r e n c e D + R so t h a t , In preferences, these but may not a pre-imposed ence o r d e r i n g i s V(F) one structures. Let that the the meaning of from expected about systematically While first DOMINANCE representation, satisfies and i n v e s t i g a t e to S T O C H A S T I C preference structures. conformed axioms a general trictions, imposed preference appealing functional start tical a AND ( 1 . 1 ) As a i n c r e a s e s The d e r i v a t i v e ^— F " = G-F i s da called s t r u c t u r e one would c o n s i d e r to impose 14 - on V i s naturally some s o r t of smoothness o f V(F ) as F s h i f t s from F to G. tt Assumption 1.1: V(F ) i s d i f f e r e n t i a b l e i n a. So f a r , we outcome simply have space which interested considered i s very probability general measures and may be non-numerical. i n monetary outcomes, we may only measures d e f i n e d on some i n t e r v a l J of the r e a l l i n e a s p e c i a l case. We denote by D 1.1 further require • n probability R, a l l o w i n g J = R as tt Consider r following: R such t h a t , ^ V ( F functional a f o r every F ) a = (l-a)F+aG, 0 -> a e [0,1], (1.2) s or the ' d i r e c t i o n a l d e r i v a t i v e ' , o f G-F (Luenberger, 1969). direction G-F i t turns and may be called a i s linear linear Gateaux or l i n e a r Gateaux u t i l i t y . out, a s u b c l a s s V can be i m p l i c i t l y therefore tt of V(F ) For a u t i l i t y 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 preference functional As J * D a n a l y s i s , " J ^ V ^ ) i - c a l l e d the 'Gateaux d i f f e r e n t i a l ' , and F * i n the d i r e c t i o n the £(•;•): a tt functional a function = /C(x;F )d[G(x)-F(x)] . C(x;F ) the 'Gateaux d e r i v a t i v e ' in I f we are that 4— V(F ) take a s p e c i f i c form. da T at some the space of such d i s t r i b u t i o n s . Assumption 1.2: For every F e D , there e x i s t s In consider on 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 the T defined defined of l i n e a r Gateaux p r e f e r e n c e by the f o l l o w i n g : - 15 - functionals J<t>(x,V(F))dF(x) = 0 , where (!>: J •*• R i s 2 functionals example i s is or linear g i v e n by the i n x and d e c r e a s i n g implicit utility in in V ( F ) . Chew This class of One such ( 1 9 8 4 ) . following: = w(x)[v(x)-V(F)]; ( 1 . 4 ) equivalently, = w(x)[v(x)-v(m)], <Kx,m) where m i s the certainty sense a ' u t i l i t y of increasing called <Kx,V(F)) (1.3) (1.5) equivalent deviation' of of d i s t r i b u t i o n F . an outcome x from the <|)(x,V(F)) certainty is in a equivalent F. Let A ( M(V\\ - o4>(x,V(F)) 4> (x,V(F)) = — • 2 It m can be v e r i f i e d f ) that - j f f i f f i f e ^ E « * . « « > = *R J * ( 1 . 6 , or ^yALV— -/<D (x,v(F))dF(x) 2 is the ( 1 . 3 ) ) Gateaux at = - derivative c ( x of ; F ) j x Y> : J the U •*• (i.7) R J linear implicit utility V (defined by F. This example expected utility. implicit utility turns out to be weighted Specifically, with the weighted preference u t i l i t y — a g e n e r a l i z a t i o n of utility functional is a subclass V being of linear explicitly given by - mrCr^ - iT(v\ V where ( F w(x) ) " is W J ( F ) /w( x)v (x)dF(x) " strictly /w(x)dF(x) positive ( ' and - called 16 - a 'weight 1 function , l - v(x) 0 v 8 ) is strictly increasing and c a l l e d With the s p e c i f i c al a 'value function'. form of V ( F ) , we can o b t a i n i t s Gateaux differenti- as f o l l o w s : a a dV(F ) da dWU(F ) da lira 4 [wu(F °)-WU(F )j a + a 9-K) 0 a ,. 1 i/vwdF ^^ l i m -5 * 9+0 /wdF 1 lim 9+0 . lim 9+0 9 /vwdF -i - /wdF I /vwdF /wdF - /vwdF /wdF L 5 /wdF /wdF a + 9 i J a 1 r 9/vwd [ G-F ] /wdF - /vwdF 9/wd [ G-F ] -5 [- 9 a a /wdF /wdF +9/wd [G-F]/wdF a g /vwd [ G-F ] /wdF /vwdF/wd [ G-F ] /wdF "/wdF a a /vwd[G-F]-WU(F ) /wd[G-F] /wdF" a /[v(x)w(x)-WU(F )w(x)]d[G(x)-F(x)] a /w(x)dF (x) a /C(x;F )d[G(x)-F(x)], (1.9) where «x;F°) g • w(x)[v(x)-WU(F )] ^ ( 1 > 1 Q ) a /w(x)dF (x) Thus, we weighted have verified utility first functional o f a l l that is linear the Gateaux differential i n the d i r e c t i o n G-F. of a Secondly, since /C(x;F)dF(x) = 0, we have implicit shown that weighted utility i s indeed utility. - 17 - a special case of linear 1.2 Frechet D i f f e r e n t i a b l e Utility Instead 1.2, of Assumption from F to G, V i s smooth i n the V(G) - V(F) = Lp(G-F) where is some l i n e a r to assuming that, u(x;F) that such L (G-F) F In other (1982a) of i n moving i.e., (1.11) f u n c t i o n a l which depends on F . V(G)-V(F) that, + ollG-Fll, to each lottery = Ju(x;F)d[G-F] = /u(x;F)dG - utilities' assumes sense of Frechet d i f f e r e n t i a b i l i t y , corresponding words, 'expected Machina can G be F using F , there is equivalent exists a function Ju(x;F)dF. approximated and This by u(x;F) (1.12) the difference as a in 'local the utility function'. Given (1.11) ^ V ( F a ) | c and ( 1 . 1 2 ) , p 0 = it can be v e r i f i e d /u(x;F)d[G(x)-F(x)]. Thus Machina's Frechet d i f f e r e n t i a b l e Gateaux utility. existence space with The present less i this supports set rest mainly spread is an L - m e t r i c ensures D ^ Q J^-J the of It linear this some the results of of than little so a l s o a s u b c l a s s of the by latter in L - n o r m on the o term i n (1.11) interval, requiring x the linear is the linear well-defined. under c o n s i d e r a t i o n say [0,M] . We denote to by restricted. develop utility expected is distributions compact essay w i l l Gateaux acceptance the general of d i s t r i b u t i o n s of utility by r e s t r i c t i n g in (1.13) on D j ( i n d u c e d spanned by D^) so that Machina those of that some r i s k and weighted utility expected u t i l i t y - 18 in aversion utility. finance Given and economics, as a benchmark for - implications the we widewill comparisons. Before ence all, ted while will a be words, the utility decision weighted utility is not a subset of on some dif ferentiable utility to will utility is non-numerical utility and both decision R. and theories outcome w i t h a l l mass centered the denote at point value x. by 6 x t i o n of random v a r i a b l e Secondly, we use and the weight , the In such c a s e s , F ~ i s ' x utility Since this outcomes, we outcomes. of In an expected functions of a step d i s t r i b u t i o n 6 stands function for the lot- to specify the random v a r i - used to denote the d i s t r i b u - x. 1 "V weakly p r e f e r r e d to G. to denote a preference For F , G i n ordering on Dj When F > G and G > F , we say F and G are following EU expected utility WU weighted utility LIU linear implicit labels to utility - 19 - (or D , "F > G" means F f e r e n t , denoted by "F ~ G " . I f F > G and not F ~ G, we say F i s p r e f e r r e d to G, denoted by "F >- G " . the expec- to be J •*• R mappings, where J Sometimes we need rr> M l ' depending on the c i r c u m s t a n c e s ) . T h i r d l y we use First Gateaux space. function In other words, D is made x t e r y of g e t t i n g x for s u r e . a distribution. on D|-Q numerical r able of implicitly on monetary with prefer- avoid c o n f u s i o n . defined focuses maker are assumed We w i l l various assumptions von Neumann-Morgenstern u t i l i t y maker of i n p a r t i c u l a r and l i n e a r perspective these representations formally defined extended finance present the clarifications Frechet can takes only other few and weighted general essay a introduce and n o t a t i o n s utility in formally theories, and terms of we indif- strictly shorten our statements: FDU Frechet d i f f e r e n t i a b l e LGU l i n e a r Gateaux u t i l i t y At times, we w i l l refer to a d e c i s i o n maker by h i s functions. For example, maker with von maker u', weight local we might refer to f u n c t i o n w as utility decision function maker with 'WU d e c i s i o n maker ( v , w ) ' , functions u(x;F) as d e c i s i o n maker whose p r e f e r e n c e 'FDU d e c i s i o n EU the are used terms expected 'decreasing', i n the weak sense. be obvious by context or so 1.3 for as decision 'EU d e c i s i o n function v and and an LGU condition (1.2) as a d e c i s i o n maker by h i s utility, WU for weighted and V for FDU and LGU. Finally, etc., value functional V satisfies preference utility, u maker u ( x ; F ) ' O r , we may i d e n t i f y i.e. utility an FDU d e c i s i o n maker with 'LGU d e c i s i o n maker C ( x ; F ) ' . functional, preference-representing an expected Neumann-Morgenstern u t i l i t y a weighted utility utility 'increasing', When the strict 'concave', sense a p p l i e s , 1 'convex , it will indicated. Representation Hypothesis EU1: There e x i s t s such t h a t , a continuous, increasing function u: J -> R for any F , G £ D j , EU(F) > EU(G) <=> F > G where EU(F) = / u ( x ) d F ( x ) . It is and o n l y i f known that the the p r e f e r e n c e (1.14) utility f u n c t i o n u i n Hypothesis ordering - satisfies 20 - the following EU1 e x i s t s axioms: if Axiom 1 (Completeness): For any F , G e D , e i t h e r F > G or G > F . j T ~ ~ Axiom 2 ( T r a n s i t i v i t y ) : For any F , G, H e D , i f F > G and G > H , then F > jT ~ ~ ~ H. Axiom 3 ( S o l v a b i l i t y ) : e x i s t s a 6 e (0,1) F o r any F , G, H e D j , such that Axiom 4 ( M o n o t o n i c i t y ) : if F > G > H , then there 8F+(1-8)H ~ G. F o r any F , G e D J } i f F > G and 1 > 8 > y > 0, then BF+(1-8)G > y F + ( l - Y ) G . Axiom 5 ( S u b s t i t u t i o n ) : F o r any F , G , H e D , and p e [ 0 , 1 ] , i f F ~ G, J then p F + ( l - p ) H ~ p G + ( l - p ) H . Moreover, any u and u* s a t i s f y i n g u* - a + b u , are e q u i v a l e n t The tively (Chew, four appealing. 1983; 'betweenness' Definition axioms a it Since F a utility sometimes 1983), and i s are called innocuous equivalent f o r any F , G e D is T curves to a p r o p e r t y c a l l e d always t r u e that F > F that a to display F > G and F between them. = ( l - a ) F + a G f o r any a e ( 0 , 1 ) of 3-outcome - 21 lines. - a the = (1- > G. any p r o b a b i l i t y mixture of i n preference i n any simplex said satisfying If that lotteries two lot- F ~ G, then F l i e s on the l i n e c o n n e c t i n g F and G, the betweenness p r o p e r t y i m p l i e s difference and norma- 'mixture-monotonicity' o r d e r i n g > on D j i s betweenness means must be i n t e r m e d i a t e ~ G. expected is A preference a)F+aG with a e ( 0 , 1 ) , ~ F 4 for a preference o r d e r i n g . 1983). betweenness p r o p e r t y i f , teries of Fishburn, (Chew, 1.1: In words, (1.15) Axiom 1984; relationship b > 0 representations first the the must segment agent's i n be straight The cause substitution for implies plex the expected that the u t i l i t y have been controversial EU to indifference parallel aimed at each be curves other. axiom. How weighted utility is the primary in d i s t r i b u t i o n . in above-mentioned the Many relaxing this and linear attempts axiom. Machina assumes F r e c h e t d i f f e r e n t i a b i l i t y to earlier to do away w i t h the it sim- generalize We mentioned theory does It will strong be e l a b o r a - shortly. The f u n c t i o n u i n (1.14) Morgenstern) utility formation that utility 1982; and u s u a l l y r e f e r r e d to Relationship the uniqueness utility theory is class an (1.15) distinct Fishburn from o t h e r d e r i v e d from a set retains the of axiomatic e (0,1), of expected is the affine trans- 1979b), Like of expected Chew (1980; expected 1981; utility, in Section 0, about the u n d e r l y i n g p r e f e r e n c e s . transitivity, theory, (von Neumann- approaches mentioned assumptions utility the generalization and Nakamura (1984). alternative completeness, axiom v i a the Axiom 5 ' (1983) as of u . advanced by Chew and MacCrimmon (1979a; 1983), axioms is function. defines Weighted is has functional agent's furthermore substitution ted preference that must axiom solvability but weakens its it It and monotonicity (strong) substitution following: (Weak S u b s t i t u t i o n ) : there e x i s t s aye For any F , G e Dj such that F ~ G and any 8 (0,1) such t h a t , for every H e D j S BF+(1-8)H ~ YG+(I-Y)H. Axiom 5' still satisfy must Definition F ~ G differs 1.2: from Axiom a relationship called For any F , G, H E D and B . F + ( 1 - B . ) H ~ 1 5 in 1 j Y-G+(1"Y-)H 1 1 - ( allowing 'ratio and 8^, - y and 8 however 1 consistency : fL,, Y]_> f o r i = 1, 22 y t B. 2, if Y2 e (0,1) such that Y /(l-Y ) y/U-y) 9 " Pj/d-pj) then, we 9 (1.16) P /(I-P ) ' 2 2 say the p r e f e r e n c e s e x h i b i t the r a t i o c o n s i s t e n c y p r o p e r t y . A proof of the f o l l o w i n g lemma appears Lemma 1.1; In x Axioms 1, 2, 4 and a simplex < x < x of l o t t e r i e s as i l l u s t r a t e d t u r e of 6 and x 6- such x imply r a t i o c o n s i s t e n c y . involving i n F i g u r e 1.1, that 6 ~P. three outcomes x, x, x e J , with suppose P i s the p r o b a b i l i t y mix- Betweenness and ratio consistency toJ x gether i m p l i e s that 'spoke out' from be to the r i g h t 5' i n Chew (1983, Lemma 2). the indifference curves must be straight a p o i n t , say A, on the l i n e c o n n e c t i n g of 6^ or to the l e f t of P, 6 x lines and P. which (A must i . e . o u t s i d e 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: strictly There e x i s t positive WU(F) > WU(G) a strictly f u n c t i o n w: <=> F increasing f u n c t i o n v: J ->• R and J •> R such t h a t , f o r any F, G a e Dj, K , where M X F X) WU(F) • Mffi $ . (1.8) f Jw(x;dF(x; The function. represent v i n (1.8) is referred Suppose another to as the v a l u e f u n c t i o n and w the weight p a i r of v a l u e and weight the same p r e f e r e n c e o r d e r i n g . the f o l l o w i n g u n i q u e n e s s - c l a s s f u n c t i o n s (v*,w*) a l s o Then v, v*, w and w* must satisfy transformation relationships: v*-f£f (1.17) w* (1.18) = (sv+t)w where q, t , r and s are c o n s t a n t s s a t i s f y i n g 23 - qt > rs and sv+t > 0. Figure 1.1: Indifference curves i n a simplex of l o t t e r i e s outcomes x < x < x - 24 - involving three Note from (1.8) that WU(F) i s not linear i n F and can be rewritten as: W WU(F) = jv(x)dF (x), (1.19) where / *w(t)dF(t) — CO = — J w F (x) . (1.20) + J °°w(t)dF(t) Clearly, when w i s constant, W WU w i l l reduce to EU s i n c e F ( x ) = F ( x ) . Because Machina's FDU a n a l y s i s not a representation theorem. approach Machina proposed as Hypothesis FDU1: There t i o n a l V: D J Q J^-J V(G) called obtain specific V(F) = /R(t)dF(t) we restate the a Fre*chet differentiable preference func- that Frechet derivative (1.21) u(x;F): [0,M] •*• R a local the implications, in ±j [ /s(t)dF(t)] assumed the follow- the 2 (1.22) ± S(x) J s ( t ) d F ( t ) . probability betweenness p r o p e r t y . that Machina f u r t h e r function = R(x) 'quadratic means comparison, form: u(x;F) with the testable with l o c a l u t i l i t y This of is f u n c t i o n at d i s t r i b u t i o n F . To ing ease approach, there - V(F) = / u ( x ; F ) d [ G - F ] + oIlG-Fll. Machina utility For not an axiomatic follows: exists -> R such is agent's 1 (1.23) functional is In a simplex of indifference curves lines. - 25 - are known to be incompatible 3-outcome l o t t e r i e s , in general not this straight Note in both that the Hypotheses outcome space i n Hypothesis EU1 and WU1 i s D^. When r e s t r i c t e d and WU are a l s o F r e c h e t d i f f e r e n t i a b l e . Hypothesis EU1 has derivative (1.21) requirement approach might not Since LGU c o n t a i n s space, 4 we s h a l l (with respect to in this structure on the ^ , of a compact FDU as tt a <x to a special essay. Later outcome space lotteries case both EU f u n c t i o n a l EU i n distributions) the Fre*chet term oilG— with means that Machina's non-compact supports. and can allow unbounded i n Section l i n e a r Gateaux d e r i v a t i v e preference 6, f u n c t i o n a l to be we w i l l need to C of outcome impose more an LGU f u n c t i o n a l i n order to h o l d under LGU. Hypothesis LGU1: There e x i s t s V(F ) that vanishes. be extendable Machina's r e s u l t s while to F u r t h e r m o r e , the adopt LGU as the most g e n e r a l discussed for constant DJQ ^ u which does not depend on d i s t r i b u t i o n F so that FII i n e x p r e s s i o n The a FDU1 i s a preference f u n c t i o n a l V: Dj •*• R such a = /C(x;F )d[G(x)-F(x)], that (1.2) v where F " = ( l - a ) F + a G and C( • ; •): We w i l l call C(x;F) the J * Bj ->• R. lottery specific (w.r.t. F) u t i l i t y function (LOSUF) of V . It should one might is not find be out them l a c k i n g known what LGU type pointed in that both FDU and LGU are structural transformation defines constraints. the uniqueness so general that For i n s t a n c e , class it for FDU or preferences. Once we i n t r o d u c e c e r t a i n t y e q u i v a l e n t r i s k a v e r s i o n , r e q u i r e d to be bounded from below. See S e c t i o n 2. - 26 - the support J is We next to examine display the the c o n d i t i o n s needed normatively appealing for each p r e f e r e n c e functional property 'stochastic called dominance'. 1.4 Stochastic Dominance It is consistent generally with stochastic Definition the agreed first 1.3: that preference dominance d e f i n e d For F, G E D degree, any F is J ( denoted by F > l Graphically, and G do not cross Definition 1.4: G, i f F(x) < G(x) a' in the ordering > is first 1 If stochas- G. degree means that F to the r i g h t of) G. said to be c o n s i s t e n t with dominance (SD) i f F > G whenever F > G. than G, if F always then delivers F ought to be a better outcome with a h i g h e r p r e f e r r e d to G. It is easy to following: Lemma 1 . 2 : a, s a i d to s t r i c t l y denoted by F > and F always l i e s below ( i . e . In other words, check the for a l l x e J . i stochastic probability dominance A preference be s a i d to s t o c h a s t i c a l l y dominate G i n f i r s t degree, stochastic should below: moreover F(x) < G(x) for some x e J , then F i s t i c a l l y dominate G i n the ordering If F > i e (0,1) Therefore, (Monotonicity) G and F " such that regardless implies = ( l - a ) F + a G , then F > i F™ > L F"' > l G for any 1 a < a . of the underlying preference tt that V ( F ) must decrease theory, in a i f V is Axiom 4 to be consis- = ( l - a ) F + a G where a e ( 0 , 1 ) , and any tent w i t h SD. Theorem U2 (SD) : F o r F , preference G £ D J F ( a f u n c t i o n a l V: D j •* R s a t i s f y i n g Assumption 1.1, - 27 - G implies if and only > F i V(F) > V(G) if G implies ^ - V C F " ) da When V i s < 0. an EU f u n c t i o n a l , d_ EU (F ) da a - lim i {EU(F a + 9 a ) - EU(F ) } 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 a {/u(x)dF + 6/u(x)d[G-F] a - Ju(x)dF } 6-K) = /u(x)d[G-F] = -Ju'(x)[G(x)-F(x)]dx , = Since F(x)-G(x) EU1, it is Ju (x)[F(x)-G(x)]dx. < 0 by the always Theorem EU2 (SD): for true that EU i s Suppose any F , G £ D j , F > > 0" "u' condition for construction preferences is of a J -*• R i s i and u ' ( x ) > 0 by Hypothesis continuous and i n c r e a s i n g . Then, EU(F) > EU(G) . referred to consistent increasing > c o n s i s t e n t with SD. G implies be of as with function, the SD. necessary Since Theroem EU2 and under sufficient EU, u stresses is that by the an EU d e c i s i o n maker are c o n s i s t e n t with SD. When V is WU(F ) i u: commonly EU to an definition a WU f u n c t i o n a l , recall that the Gateaux differential of is a dWU(F ) = da a JC(x;F )d[G(x)-F(x)] (1.9) a = -/[G(x)-F(x)]dC(x;F ) = -/C (x;F )[G(x)-F(x)]dx. , (by i n t e g r a t i o n by (1.24) a - 28 parts) - i If ssary F > G , then G ( x ) - F ( x ) and s u f f i c i e n t > 0 f o r a l l x e J by d e f i n i t i o n . condition for : da J for < 0 is therefore The neceC'(x:F) >0 a l l x e J at a l l F e D j . Theorem WU2 (SD): Suppose increasing strictly positive. and bounded; v i s Then, strictly for any F , G £ D j , l F if and w i s w, v are continuous > G i m p l i e s WU(F) > WU(G) and only i f C(x;F) = w ( x ) [ v ( x ) - W U ( F ) ] / / w d F is an i n c r e a s i n g or, function (1.25) of x for a l l F e D j j equivalently, <t>(x,s) = w ( x ) [ v ( x ) - v ( s ) ] is an i n c r e a s i n g Confirming f u n c t i o n of x f o r a l l s £ J . Theorem U2, Gateaux d i f f e r e n t i a l that, under (1.26) the condition for of WU(F) be d e c r e a s i n g EU where u'(x) a l s o has the f u n c t i o n a l > 0 guarantees analytical SD under WU i s that i n the d i r e c t i o n G - F . its c o n s i s t e n c y with i n t e r p r e t a t i o n of a Gateaux the Recall SD, u(x) derivative, but does not depend on the d i s t r i b u t i o n F because the EU r e p r e s e n t a t i o n EU(F) = Ju(x)dF(x) is linear name the C(x;F) use of such, the in distribution. a 'lottery this Machina's term 'local In 'local' risk specific term to 'local here aversion', This observation utility l e d Chew and MacCrimmon to function the Gateaux d e r i v a t i v e utility since function' its meaning to be d i s c u s s e d Theorem WU2, the condition (LOSUF)' . is also is different SD i s apply of an LGU f u n c t i o n a l . i n the next for We a l s o a LOSUF. We avoid from the As using 'local' in section. given in terms of both C(x;F): J x Dj -*• R and <t>(x,s): J + R. z and conditions on tion-free, i t may at times o f f e r more i n t u i t i v e C(x;F) ())(x,s) are Since equivalent. The c o n d i t i o n for SD i n terms of ^(x.s) = w'(x)[v(x)-v(s)] which i n t u r n can be r e w r i t t e n w'(x) / s w(x; > , < When [ l n w ( x ) ] ' f [lnw(x)]' Let In other are of tlnw(x)]' words, and from when w(x) decreasing, In the is to r is a 1 as: > 0, (1.27) < s * x. > 1 /. (1.28) o equivalent x o N to: for a l l C being V below is by ( x ) < ' v-v(x; > - y' , v(x)-v SD r e q u i r e s above, concerned "weighted distribu- interpretations. s $ x. > (1.29) C o n d i t i o n (1.29) can then be ( the the x ) rate of RHS of increasing, i f w'(x) > 0, (1.30) i f w'(x) < 0. (1.31) change of and cannot increase lnw(x) (1.31), F will only one be preferred to G as from respectively too fast; — when w(x) either. we c o n s i d e r e d m u l t i p l e d i s t r i b u t i o n s . that with lnw(x) be bounded (1.30) lnw(x) cannot decrease too r a p i d l y guarantee Note o /s j v v(x;-v(s) = [lnw(x)]' w(x) is + w(x)v'(x) is given F , as ^ r - = i.e., ( - I < <|)(x,s) <|>(x,s) can be w r i t t e n v = max | v ( x ) } and v = min (v(x) }. rewritten above (1.28) As constant as: v'(x) , 7—T 7—v v(x;-v(s; exists, /wdF i s long particular distribution, The SD c o n d i t i o n as F > what is l G. the If we meaning increasing? from e x p r e s s i o n (1.25) utility-deviation that C(x;F) from WU(F)" with - 30 - has the interpretation w(x)//wdF being the of a weight. It is t h e r e f o r e not , , . _ C (x,F) = 7wof I [ v ( x ) - a c c o r d i n g l y a "marginal increase in combined effect that /C(x;F)dF(x) The d e r i v a t i v e C(x;F) caused of two W U ( F ) ] + V 7wlF- ' weighted by ( x ) ( an i n f i n i t e s i m a l forces represented by increase of the terms two x, are constant. w(x) and v(x) positive. (v,w) o p t i m i s t i c or p e s s i m i s t i c is creases, Second, first it it term w'(x) may be viewed strictly causes changes gives may be p o s i t i v e two e f f e c t s the the on £ . change of According First, 'sure it in the utility, positive assuming i t s b l e cases as l i s t e d case effect C resulting from v(x)-WU(F) has not changed. g l ^ g l [ v ( , ) - w P ( F ) ] v'(x) to (1.32), the weight v(x). the In is the change in level. The second increase weight, i n the c o n t i n - There are four possi- ^ v ' ( x ) C'(x;F) + + + + (2) + - - + ? (3) - + - + ? (4) - - + + + - the w(x). (1.32), + 31 the when x i n - (1) - ) (1.32). below: ^ 2 depending on whether initial on C caused by the weight 3 WU(F) and unchanged. changes utility' h o l d i n g the c o n t i n g e n t u t i l i t y - d e v i a t i o n at i t s term gives as or n e g a t i v e at x. contingent - and i s a d i s t r i b u t i o n F , suppose x i n c r e a s e s m a r g i n a l l y to x . /wdF 1 u t i l i t y - d e v i a t i o n from WU(F)" — + Given gent = 0. w'(x)[v(x)-WU(F)]+w(x)v'(x) r is surprising Obviously, > 0 i n both sively case tainty when x i s pose no ambiguity. (3) requires better that than h i s 'overly-optimistic' To a l s o have the agent certainty either utility repeatedly is C(x;F) function give u(x) . see the If later of the For affine i n the this is is as far of be not equivalent when x i s as of below the preference the cer- properties This transformation u(x) is must a l s o be obviously although it is the Gateaux G implies and only if C(x;F) is Given true rarely in so derivative a of V ( F ) at F i s C(x;F), the true: with LOSUF C: R * D j •*• R. F able literature. Theorem LGU2 (SD): Let V: Dj •*• R be a l i n e a r Gateaux p r e f e r e n c e that exces- the von Neumann-Morgenstern to make sense, interpretation. EU uniqueness LGU, since following that, WU e q u i v a l e n t a utility-deviation interpreted if C'(x;F) equivalent. concerned, light and (4) and case and not We w i l l to (1) (2) pessimistic distribution, are cases the satisfy Then, V(F) > V(G) increasing plausibility the functional required i n x for a l l F e D j . of SD, we will conditions. - 32 - consider only the functions 2 INDIVIDUAL RISK AVERSION In f i n a n c i a l economics, averse. The n o t i o n based on d i f f e r e n t lent of we may say that Alternatively, risk if always he is risk We can even his risk is risk attitudes do clearly hold risk that 'sizes'. his in risks risk sense. agent have willing to pay than the mean of that to risk, i n the sense o f i n s u r a n c e premium. of of an a g e n t ' s asset demand or and interest to d i s t i n g u i s h large'. The o b e r s e r v a t i o n lottery different position, towards tickets attitudes termed risks risk local i n general agents' that simultaneously towards an a g e n t ' s conventionally aversion is etc. policies is an a g e n t ' s c e r t a i n t y e q u i v a - i n the c e r t a i n t y e q u i v a l e n t and ' i n the people differently lotte- terms often Given a wealth infinitesimal contrast, small' insurance suggests different wards ' i n the averse information, is be d e f i n e d risk v a l u e of that always g r e a t e r risk it if premium an aversion another d i r e c t i o n , people aversion risk is risks of to- aversion. called global aversion. Under tions is expected are a l l some of it define he s u b j e c t i v e v a l u e of In In that is d e c i s i o n makers are than the expected averse insurance t r a d e away an a r b i t r a r y r i s k then we may say less that can however For example, is the assume aversion concepts. for any l o t t e r y ry, we o f t e n these utility, equivalent risk necessary to aversion that however, the concavity notions we make these are not distinctions - of 33 - different the risk utility equivalent between aversion function. no- Since under WU and LGU, them. We w i l l begin with l o c a l r i s k 2.1 aversion. Local Risk Aversion In risk the literature a v e r s i o n towards of risk small r i s k s . mal , a c t u a r i a l l y f a i r r i s k the d e c i s i o n maker always >^ F ^ ) > we would x + a v e r s i o n at x. mal, pay it to insure than E ( e ) , the it small r i s k define risk e is aversion say refers an a r b i t r a r y and the d e c i s i o n maker's wealth that his preferences level against seems is this risk. reasonable As display local long as the this If 6^ risk infinitesi- * 0 and the to say that x. e (i.e. r i s k being c o n s i d e r e d here has to be Suppose E( e) to infinitesi- s t a t u s quo to t a k i n g r i s k need not be a c t u a r i a l l y f a i r . greater To like local Suppose prefers his While the a premium to towards aversion, agent can premium agent is is averse e. local risk aversion formally, we first define the term ' i n s u r a n c e premium'. Definition ; 2.1: 6 _/~\ , x+E(e;-ii risk e at We a l s o If then a decision % is maker called his is indifferent (unconditional) and p6 v define 2.2: mium for r i s k ry r i s k If x+E(e)-n First, i n s u r a n c e premium for c x. 'conditional insurance f o r any n o n l i n e a r - i n - d i s t r i b u t i o n p r e f e r e n c e Definition between F , ~ and x+e a decision +(l-p)H, ' K then maker i s % is premium' which w i l l be needed theories such as WU and LGU. indifferent between p F ^ + ( l - p ) H called his x + c o n d i t i o n a l insurance p r e - e at x c o n d i t i o n a l on p and H. note that which need not the be risk e in Definitions actuarially - 2.1 and 2.2 is f a i r or i n f i n i t e s i m a l . 34 - an a r b i t r a Second, the x in Definitions sure wealth it(x,e), 2.3: can be to taking 2.4: risk a decision then maker's his e, If a decision Local n(x, £jp,H), upon, risk restricts Definition play all then is his aversion If e, we say Clearly, positive tudes. terms aversion that it Pratt LRA. that that s a i d to d i s p l a y 2.6: the the agent's to insurance to r: of display finitesimal £ with variance conditional the IPRA because infinitesimal premium f o r any ones. infinite- preference is said preference displays to dis- LRA at define a term c a l l e d in this essay. £ and the a g e n t ' s r i s k atti- a. How can we express function? The now famous 1 n in Arrow- way. (unconditional) r(x+E(£)) is LRA (PLRA). £ is J -> R i s the premium f o r and H £ D j i t unconditional his on x, utility ence o r d e r i n g i f TI(X,£) = ^ of If (uncondi- x. We c o u l d c o n c e i v a b l y variance A function risk x. e, x. then h i s pointwise index p r o v i d e s a convenient Definition said insurance n depends u will risk. T h i s however w i l l not be c o n s i d e r e d previously E , X and (LRA) at displays IPRA i m p l i e s Assume of is case positive, maker's general, f o r any p £ (0,1] a special is decision In towards is (CIPRA) at maker's the e. conditional preference a decision ' c o n d i t i o n a l PLRA'. said is 7t(x,£), local risk We risk preference the r i s k s under c o n s i d e r a t i o n 2.5: risk x, as i n s u r a n c e premium f o r any r i s k maker's i n s u r a n c e premium r i s k a v e r s i o n simal the individual's attitudes positive, conditional it interpreted i n s u r a n c e premium r i s k a v e r s i o n (IPRA) at Definition any If 2.2 prior E , and the is tional) and position depend on x, Definition 2.1 a + o(a ). z z an A r r o w - P r a t t index of i n s u r a n c e premium T t ( x , £ ) •*• 0 can be w r i t t e n a preferf o r an i n - as (2.1) Given D e f i n i t i o n 2 . 5 , Corollary 2.1: and only i f The r(x) > 0. following with He i s theorems PLRA i f Index): a continuous, = - shown t h a t , strictly LRA at x i f > 0 at a l l x e J . index increasing, f u n c t i o n u(x) The p r e f e r e n c e is of an EU d e c i s i o n twice-differentiable of actuarially fair at wealth p o s i t i o n g i v e n by an EU d e c i s i o n maker u w i l l let risk (2.2) 7 display it be h i s i n s u r a n c e premium f o r an e with small variance a . 2 It can be x, z r r Theorem WU3 ( A r r o w - P r a t t von concave. / ~\ ° v"(x) u(x,e) - - - j . [ _ y - H - r(x) The A r r o w - P r a t t a WU d e c i s i o n maker ( v , w ) , infinitesimal, and o n l y i f is v PLRA i f and only i f u i s For index r ( x ) • u'(x) Theorem EU4 (PLRA): obvious. are w e l l known: Neumann-Morgenstern u t i l i t y r(x) below i s A d e c i s i o n maker w i t h A r r o w - P r a t t Theorem EU3 ( A r r o w - P r a t t maker C o r o l l a r y 2.1 + , 2w'(x), _^i.] Index): , + 0 „ •>* <^). ,„ „. (2.3) The A r r o w - P r a t t index of a WU d e c i s i o n maker w i t h p r o p e r l y s t r u c t u r e d v a l u e f u n c t i o n v and weight f u n c t i o n w i s g i v e n by r(x) = It is [l£E> + v'(x) worth n o t i n g WU A r r o w - P r a t t Expression risks can J that, = - ^4 v'(x) like i n (2.4) its is - ^ L . w(x) expected v utility (2.4) ' counterpart, i n v a r i a n t under the uniqueness the class (1.17) and ( 1 . 1 8 ) . (2.4) be suggests seen a d d i t i v e terms. The f i r s t aversion which index' ( x ) ; ] w(x) index r ( x ) transformations small 2 w L as that a WU d e c i s i o n maker's a v e r s i o n toward coming from term - v " / v ' measures risk - two sources represented can be c a l l e d the aversion 36 - by 'value-based attributable to the two risk value function v. The second term -2w'/w can be i n t e r p r e t e d as the based r i s k a v e r s i o n index' reflects certain prospects two in question components neither qualities must necessary constant, or simply the of the (Weber, be nor positive at sufficient EU r e s u l t s , 'optimism (pessimism) decision 1982). maker's To d i s p l a y all for x. r(x) reduces the preference PLRA, When the to the index' perception the The c o n c a v i t y PLRA. and 'perceptionthat about sum of of weight traditional the these v alone is function is Arrow-Pratt index. By Corollary display LRA at 2.1, x if and o n l y if of r(x) a WU d e c i s i o n given by (2.4) is maker (v,w) will positive. Given that r ( x = - ) [1^0 v (x) + 2wJWj w(x) ' L ( 2 2 = - { l n [ v ' ( x ) w ( x ) ] }', the following is Under vered E U , the Therefore, 2 ln[v'(x)w (x)] is index r ( x ) will two EU maximizers function can be u n i q u e l y (up to an a f f i n e reco- transfor- pairs (2.6) who share the same Arrow-Pratt index must function. above, from the A r r o w - P r a t t distinct dis- decreasing i n x. = /exp[-/r(x)dx]dx. From ( 2 . 5 ) Pratt a WU d e c i s i o n maker (v,w) following: have the same u t i l i t y two of von Neumann-Morgenstern u t i l i t y from the A r r o w - P r a t t u(x) ) (2.5) The p r e f e r e n c e and o n l y i f mation) v i a the 4 obvious: Theorem WU4 (PLRA): p l a y PLRA i f . it index of index and e x h i b i t is is value clear that, under WU, what we can r e c a p t u r e 2 v'(x)w (x). and Therefore, i t weight functions identical local risk - 37 - share propensities. is the possible same that Arrow- The and conditions sufficient. tions i n Theorems EU4 and WU4 f o r It will are s u f f i c i e n t fies PLRA are both necessary be i n t e r e s t i n g . to know what more s p e c i f i c f o r WU to d i s p l a y PLRA. Corollary 2.2 condi- below identi- will display two such c o n d i t i o n s . Corollary 2.2: PLRA i f The p r e f e r e n c e condition (i) of or ( i i ) (i) w is constant and v i s (ii) v is l i n e a r and w i s a WU d e c i s i o n maker (v,w) below h o l d s : concave; decreasing. Proof: Omitted. In case b e i n g concave (i) is condition ( i i ) and where w i s necessary where v i s a decreasing w w i l l constant, WU reduces and s u f f i c i e n t linear, result to EU. Consequently, f o r PLRA by Theorem EU4. the f i r s t in a positive term of r ( x ) Arrow-Pratt i n (2.4) Arrow-Pratt risk with index. small Suppose 2 variance o . E is Under vanishes index. To c h a r a c t e r i z e LRA f o r an LGU d e c i s i o n maker V , we must f i r s t his derive 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 Then, by the definition of fair insurance premium, V Let F a < W = (l-a)F = V F < X+E>- ~ + a6 X+E . We have X-1X 0 = V( 6 J x-ir - V(F ~) = X+E a J t = /J{/C(s;F )d[6 _ = /{/JcCsjFVaJd! C ^ - F = /C(s;F ')d[6 _ x a x 7 t i i - F - F x + x + da da 0 ~]}da ~] x + (from ( 1 . 2 ) ) ~] f o r some a' Hence, - 38 - v £ (0,1). C(x-ii;F ') a = /C(x+s;F ')dF~. a OC' ^ N o t i n g that Taylor's e is a small r i s k and that F expansion f o r both s i d e s as C(x-n;F ') a - C(x-7t;6 ') •> § x as o z + 0, we can take the follows: ) = C(x; 6 ) - it C (x; 6 ) + 0 ( i t ) ; 2 and / C ( x + s ; F ' ) d F ~ - /C(x+s;6 )dFa = J[C(x;6 ) + sC'(x;6 ) + 1^ C " ( x ; 6 ) = C(x;6 ) + o(a ). x x +^ x C"(x;6 ) x + x 2 o(s )]dF; 2 Therefore, C"(x;6 ) . ] + o(a ) > x 2 it = it(x, e ) = _ [ - 2 x CT a 2 r(x)+ o(a ), 2 = 5 (2.7) where C*(x;6 ) x r ( x ) " C'(x;6 ) = x When E ( e ) * 0, * ( expression „ v 7t = Tt(x, e) = ^ [- (2.7) , 8 ) becomes M C (xt-E(e);6 * C'(x E(e);6 + 2 ~) + E { x + E ( e ) ~ ) 2 ] 4- o ( o ) . (2.9) ) Hence, Theorem LGU3 ( A r r o w - P r a t t I n d e x ) : maker V w i t h c o n t i n u o u s , C(x;F) is C(x;F) Note 2 ln[v'w ] will (PLRA): the (2.8) The p r e f e r e n c e d i s p l a y PLRA i f that, of of u an LGU d e c i s i o n maker' V with LOSUF concave i n x f o r a l l F . being concave in Theorem i n Theorem WU4 are both n e c e s s a r y - LOSUF above. C(x;F) i s conditions being d e c r e a s i n g index of an LGU d e c i s i o n s t r i c t l y increasing, twice-differentiable g i v e n by e x p r e s s i o n Theor em LGU4 The A r r o w - P r a t t 39 - and EU4 and sufficient while 2.2 the concavity C(x;F) i n Theorem LGU4 i s risk same and in G has the large. If a higher sufficient world, it the is for any aversion. It nonlinear prevalence of various f o r PLRA. unfair is, desirable a v e r s i o n when a s p e c i f i c to definition simpler known v i a of then forms risks we of — would to display (1970), such a 1 spread, concept we theory, throughout for which we c a l l with risk of global be i t the preferrisk l i n e a r or some form of g l o b a l Instead start a the concept. risk it. global r i s k aversion in 'mean p r e s e r v i n g s p r e a d . mean p r e s e r v i n g expect an i m p l i c a t i o n a p p l i c a t i o n context c a l l s and more i n t u i t i v e F and G share lotteries for a u t i l i t y be a b l e A f t e r R o t h s c h i l d and S t i g l i t z better distributions (GRA) r e f e r s to r e q u i r e a l l agents not to have actuarially in distribution, two Global r i s k aversion is perhaps u n r e a l i s t i c however, global r i s k aversion variability, agent to p r e f e r F to G. Given ence to l o c a l r i s k a v e r s i o n , aversion mean averse is only Global Risk Aversion In c o n t r a s t to of of finance jumping i n t o a less 'simple general, mean the but preserving spread'. D e f i n i t i o n 2.7: For F if G , G i s s a i d to s i n g l e - c r o s s F at x* from the left if G(x)-F(x) > 0 f o r a l l x < x* (2.10) G(x)-F(x) < 0 f o r a l l x > x*. (2.11) and When there possess the is no single ambiguity about the direction, crossing property. - 40 - we say that G and F D e f i n i t i o n 2.8: G i s a simple mean p r e s e r v i n g spread ( s i m p l e mps) of F i f (a) G s i n g l e - c r o s s e s 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, c o n d i t i o n identical; condition (a) i m p l i e s (b) i m p l i e s that a mean-variance type agent, F c l e a r l y ing requirement F. To see t h i s , d i s t r i b u t i o n s with the same mean. F from the l e f t The mean p r e s e r v i n g spread than F. The s i n g l e c r o s s suppose F, G and That H s i n g l e - c r o s s e s does not imply defined variability dominates G. i s however not t r a n s i t i v e . G single-crosses the mean of G and F i s G has a g r e a t e r For H are three that that H w i l l G and single-cross below v i a second-degree stochastic dominance i s l e s s r e s t r i c t i v e but t r a n s i t i v e . Definition the 2.9: F o r F, G e D j , F i s s a i d to s t o c h a s t i c a l l y dominate G i n second degree, denoted by F > T(y) = lj„ [ G ( x ) - F ( x ) ] d x z G, i f > 0 fora l l y e J, (2.13) and T(«) = / [G(x)-F(x)] J = £ JG(x)-F(x)]dx = 0. A l t e r n a t i v e l y , G i s s a i d to be a mean p r e s e r v i n g When the means of F and G e x i s t , c o n d i t i o n are equal. their all Condition (2.13), 'squeezed' means — the mass over greater spread (mps) of F. (2.14) i m p l i e s represents m This y, then the squeezed that they a requirement i f we a r b i t r a r i l y p i c k a p o i n t y and [ y , ) onto than that o f F. i n contrast, (2.14) on concentrate mean of G must not be can be seen by r e w r i t i n g c o n d i t i o n (2.13) as (2.15) below: T(y) = |/JxdF(x)+y[l-F(y)] Condition } - { / J x d G ( x ) + y [ l - G ( y ) ] } > 0 f o r a l l y. (2.13) w i l l o b t a i n (2.15) i f F and G have the s i n g l e c r o s s i n g p r o - 41 - perty and s a t i s f y the equal mean c o n d i t i o n fore a special case of mps. In f a c t , that an mps of F can be viewed as a r e s u l t (2.14). Simple mps i s R o t h s c h i l d and S t i g l i t z there- (1970) show of a sequence of simple mps' of F. D e f i n i t i o n 2.10; aversion (MRA) at preference all is F if said to he always display prefers is F to s a i d to d i s p l a y mps r i s k G whenever g l o b a l MRA (GMRA) if it F > 2 G. displays His MRA at F. Lemma 2 . 1 : a, A d e c i s i o n maker's p r e f e r e n c e a' If F > e (0,1) Hence, tonicity) 2 G and F such that regardless = (l-oc)F+aG, then F > 2 F t t > the u n d e r l y i n g p r e f e r e n c e 2 G f o r any t h e o r y , Axiom 4 (Mono- that V ( F ) must decrease i n a i f V i s (GMRA): > F"' 2 a < a'. a implies Theorem U 5 . 1 of t t For F , G e D , F to d i s p l a y GMRA. = ( l - a ) F + a G where a e ( 0 , 1 ) , a and «J any p r e f e r e n c e F if > 2 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, G i m p l i e s V ( F ) > V(G) and o n l y i f F > 2 G implies ^ Another way of tainty < 0. characterizing risk a v e r s i o n i n the large is via cer- equivalent. D e f i n i t i o n 2.11: c is tt V(F ) I f a d e c i s i o n maker i s s a i d to be h i s Definition 2.12: If (unconditional) a decision c certainty equivalent (CCE) of F, indifferent then c i s 42 - then (CE) of F . between two compound s a i d to be h i s conditional 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 t r i b u t i o n H. - between F and 6 , c certainty equivalent maker i s l o t t e r i e s p F + ( l - p ) H and p 6 + ( l - p ) H , indifferent dis- Definition 2 » 1 3 : If preference aversion is Definition always to said (CERA) (GCERA) i f a d e c i s i o n maker always it to at display F. His displays 2.14: (unconditional) preference is to preference + ( 1 - p ) ^ y j p P H to his equivalent said if ( l - p ) F + p H , then h i s X( is F> then display F o r any F , H E D J and p e ( 0 , 1 ] , prefers 0 X ( certainty said t °y jp risk global CERA CERA at a l l F . display conditional certainty His prefers equivalent to display (2.14) implies global a d e c i s i o n maker preference r i s k aversion CCERA (GCCERA) is said (CCERA) at F . if it displays CCERA at a l l F . Definition (GCCERA), expected 2.13 then his value of axiom r e q u i r e s p, it CE (CCE) of that that if cal. axiom i m p l i e s any if a lottery is Under expected G ~ F , then f o r must be t r u e that tution pG+(l-p)H ~ p F + ( l - p ) H . that the CE and CCE of simply r e q u i r e s and H. As utility, that such, if CCERA i s p6 +(l-p)H c an a g e n t ' s decision always maker is smaller utility, the GCERA than the substitution 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 ~ F , the substi- any d i s t r i b u t i o n are identi- T h e r e f o r e , CERA and CCERA are e q u i v a l e n t Beyond expected It lottery. that Since 6 c under expected utility. weaker than the s u b s t i t u t i o n axiom. ~ p F + ( l - p ) H , then c < /xdF f o r a l l p CE and CCE of a d i s t r i b u t i o n F need not be he must be equal. A paid decision to premium context, that give given up in agents he has maker's a CE can be lottery Definition can be with 2.3 viewed interpreted positive is as expected a form of seeking been endowed with a r i s k - of 43 as to the amount value. CE s i n c e , sell adverse n e g a t i v e expected - in The insurance an insurance risks value, — Given how much would he be also insurance use (IPRA and willing to pay f o r t r a d i n g away t h i s premia to CIPRA are d e f i n e d Definition 2.15: A (unconditional) characterize risk? global i n D e f i n i t i o n s 2.3 and d e c i s i o n maker's preference Therefore, risk 2.4, aversion we as can below respectively): i s said to d i s p l a y global (GIPRA) i f i t d i s p l a y s IPRA at a l l wealth levels d e c i s i o n maker's p r e f e r e n c e i s said to global IPRA x. Definition 2.16: A display c o n d i t i o n a l IPRA (GCIPRA) i f i t d i s p l a y s CIPRA at a l l wealth l e v e l s x. As CERA draw only Now and IPRA the one that we have introduced A preference (CD) >1 V(G) CCERA and CIPRA), we will i s s u e under d i s c u s s i o n . following property ordering of to impose lower >: > i s s a i d to d i s p l a y c o n t i n u i t y i n G and the sequence {G^} converges e x i s t s an N > 0 such that f o r every n > N, F V Suppose 0, there are GCCERA, i t i s necessary i f whenever F >- to G i n d i s t r i b u t i o n , (so to the C o n s i d e r the 2.17: distribution equivalent more r e l e v a n t boundedness on J . Definition are f u n c t i o n a l V represents preference have compactsupports, CD means that if \G 1 \ ™ n n=0 converges J worse-than s e t ' i s c l o s e d , i m p l i e s Axiom 3 In below. order To converges or G. |V(G )} ™ n Graphically, the 'better-than CD n 0 >. will When G^, converge to means that the s e t ' i s open. n = 'not- Clearly, CD (Solvability). for GCCERA illustrate, 0,» qH e (0,11. \ > J to ordering SD and and CD to consider x + compatible, x+ 6 Q,. >X + must bounded from 44 - x 6x ~> F q f o r q e (0,0.5], g as q -> 0, - J = (1-q) 6 g + q & _ j Q/q)_]_ j Q> where 9 > F^ GCCERA imply f J i n d i s t r i b u t i o n to 6 be c o n t r a d i c t i n g CD. but In the Fq re- mainder of this ever GCCERA. i s So f a r , aversion, i.e. essay, assume that J is bounded from below when- involved. we have given four d e f i n i t i o n s of PLRA, GMRA, GCERA and GIPRA, and two i.e. GCCERA we w i l l and GCIPRA. The following is unconditional global of obvious risk c o n d i t i o n a l ones, in light of their definitions: C o r o l l a r y 2.3: GCCERA + GCERA -»• PLRA. We have a l s o p o i n t e d out GCCERA and GCIPRA). It turns underlying Beyond t h i s , that, preference equivalence monetary out for that theories. what follows, then are of the together? equivalent and (so regardless we first extend it prove to this arbitrary lotteries. A l o t t e r y of the form s A i o = -^-6 i = l N x_^ N denoted by x = other words, an elementary comes x^ , not outcomes In lotteries' c a l l e d an elementary l o t t e r y , need how are they l i n k e d GCCERA and GMRA are 'elementary D e f i n i t i o n 2.18: In GCERA and GIPRA are e q u i v a l e n t x^ w i t h be uniform distinct. with Thus, rational lottery is lottery probabilities can ^ is (x^,x ,•••,x^). 2 a l o t t e r y which g i v e s N o u t - p r o b a b i l i t y 1/N. any + . . . + -U Note that x^, involving be a expressed x^ finite number as elementary an of lottery. The following is due to Hardy, L i t t l e w o o d and P o y l a (1934): N Definition 2.19: For v e c t o r s m a j o r i z e s y j , denoted by x > (a) ' v and n 2. ,x. i=l i > n E. ,y, i=l i - , x, m £ £, e J , x is a m a j o r i z a t i o n of £ (or x if f °r a 1 1 1 < n < N (2.16) (b) E f x where When the In mean. = j L E ^ y . elements inequality majorize £ the 1 i of (2.17) x and (2.16) y_ have holds been strictly arranged i n ascending f o r at least one n, x i s order. said to strictly. D e f i n i t i o n 2.19, There i s mean , condition (b) implies a sense i n c o n d i t i o n (a) than £ when x m a j o r i z e s £. given i n D e f i n i t i o n 2.9. equivalent the to second-degree that x i s This s e r v i n g spreads that more sounds As i t x and £ share the 'centered' s i m i l a r to same towards the mean p r e - turns o u t , majorization stochastic dominance for m and o n l y i f x > is elementary lotteries. Lemma 2.2: Proof: For elementary Express lotteries, x > £ if N N x and y as F = E . , -JH5 ~ ^ 1=1 N x. l and G = S . . -^-6 , i=l N y ' i 2 £. respectively. v J J Since = \ JIJHIGCX) c o n d i t i o n (2.14) (Sufficiency) that x^ > y ^ . rality, J = y^. X = L = E . = 1 implies We prove Suppose = - = - y ± the X E i = i y i , i m p l i e d by e q u a l i t y by i n d u c t i o n as J (2.17). follows. c o n t r a r y that x^ < y^. < y, n E N i=l i' and i s (2.16) a l s o assume x ' N i n+1' we show Without l o s s of < x , , , where n = 1, 1 First, or N. ' gene- Consider z ' Then, fz , „ , > . r i ri/ M J ^ x d F ^ ) + z[l-F(z)] 1 ~ n = _ E x. . 1 < < This contradicts (2.15), . N—n 1 + _ z = ^ i = 1 , n-1 x i + y l + N 1 -r + -r = a condition for F > - 46 - , „ n + E. x. . N—n + — z = 2 , N-n z z T X ; L z ^-« 2 G. x d G ( x ) + Z 1 G Z [ " ( )]- T h e r e f o r e , x^ > y ^ . We next assume t h a t , f o r some k < E x i=i i > E N, ( 2 a 8 ) y i=i r k+l It remains to be ^ x shown t h a t < ^ y ± Inequalities ± k+1 x > ^_^ ^ y S u ^i=l i" PP o s e the c o n t r a r y - (2.18) and (2.19) (2.19) together x without l o s s of g e n e r a l i t y that ^ ^ < + J!,xdF(x) + z [ l - F ( z ) ] - I x i y i + ^ ± < k+l* X < k+i n S ; x y imply that * n+1* 2 j X X ,X, + v , 1 Jc+1 < N = S Z , y it contradicts £,xdG(x) + follows that E. > i=l i n i X j E^y. i=l l (2.15). = y s u m e k+l° N-n N Z Z z[l-G(z)]. k+l again Z s N-k-1 "IT" + i=l i L e t A z 1 k+l 1 n N i=l i N i=k+2 i . 1 k+l . N-k-1 < — E. N 1=1 i N This that Jc+1 Hence, ^ _ ^ x ^ > ^_^y^* f o r a l l 1 < n < N. By induction, (2.16) ( N e c e s s i t y ) For any z e J, suppose without l o s s of g e n e r a l i t y that X < z < x , and y, < z < y, ., . There are three cases to c o n s i d e r : ( i ) k = n+1 k ^k+1 n, ( i i ) k < n, ( i i i ) k > n. R Case ( i ) : k = n J!„*dF(x) + = ± \l^ + ^ ± . 1 „ n " N i=i i z = y + , N-n ~ir J_ xdG(x) + o ' z z[l-G(z)] Case ( i i ) : k < n Jf^dFCx) + z [ l - F ( z ) ] = I - E.^x 47 + 2=» 1 - z , > 1 „ n N i=l i S y + 1 „ k N i=l i E = . > y y + z , l „ n N i=k+l i + 1 „ k N i=l i S , N-n T " Z y , N-k — , N-n T + Z Z = jf^xdGCx) + z [ l - G ( z ) ] . Case ( i i i ) : k > n J f J«1F<X) + z [ l - F ( z ) ] = 1 E ^x i . > 1 „ n N i=i i s y .1 „ n N i=l + 3jp 1 + k-n i r = E N z N-k - r + z . 1 „ k N i=n+l x N-k N J i 1 _ k = z N-k y i=i i + z -r jf^xdGCx) + z [ l - G ( z ) ] . Q.E.D. We proved Lemma 2.2 are It equivalent is easy lotteries (2.15) fied is For zed the check is of mean' that not as if direct. algebra. the c o n d i t i o n s second-degree (2.14) (2.17) That That is for m a j o r i z a t i o n stochastic satisfied holds. (2.16) (2.16) for elementary The e q u i v a l e n c e implies is (2.15) also between is veri- necessary c o n d i t i o n (2.15) i m p l i e s that the p r o b a b i l i t y measure on [z,x^] i s the for In c o n t r a s t , x (i.e., condition the mean of (2.16) says that a reduced vector - 48 - the 'z-squee- squeezed where z < x^) must not be s m a l l e r than the l i k e w i s e of dominance. proved by i n d u c t i o n . elementary l o t t e r i e s , z, for condition and o n l y straightforward mean' of x ( i . e . , point conditions x and £ i f and (2.16) via (2.15) to to by v e r i f y i n g that to the squeezed mean 'n-element (x^,...,x ) n partial with n < N) must not be original to that ginal than that of Note that a 'squeezed lottery' l o t t e r y with the r i g h t t a i l beyond a f i x e d p o i n t being p o i n t while one with servation and less a 'partial lottery' uniform ( c o n d i t i o n a l ) is a truncated l o t t e r y p r o b a b i l i t y 1/n. is 'squeezed' of the ori- The f o l l o w i n g should p r o v i d e more i n t u i t i o n f o r the e q u i v a l e n c e the between ob- (2.15) (2.16) . C o r o l l a r y 2.4: jn F o r elementary jn i ,m > x. x ... > l o t t e r i e s x, k jn £ jn > ... £ e , x > m £ implies N-l _ > = £ y_> where X - Z l ( X 1 X } , X 2 x " (yi' 2 , 3 + ( x . . . , X y ) , X n n x 2 N_1 x, n + 2 , . . . X . X ^ . X j j ) , X n' n+l n+2' , , , , X , X N-l N +E ( ) n - V i i - i V i •V 2 » • • • •V i y y x 1 y y y = (yi.y ' 3'---' n' n+i'yn+i'---»yN-i' N ) P r o o f : Omitted s i n c e Corollary (2.17), ed it 2.4 is means a finite 1 y^ , that, if again of push x ^-y^. z 2 ) ) then £ can be o b t a i n e d < . leftwards this to y same d i s t a n c e - 49 of to a p o s i t i o n - 2 obtain- rightwards To o b t a i n £ 2 (by a d i s t a n c e 1 y^ i s pushing x new p o s i t i o n as z . 2 From (2.16) and S t a r t i n g from x, to y^ and s i m u l t a n e o u s l y Label p u s h X g r i g h t w a r d s by the x majorizes of mps' or m a j o r i z a t i o n s . we know that x^ > y^ and by a d i s t a n c e y r i straightforward. sequence by p u s h i n g x^ l e f t w a r d s ( x (2 2 from [ x + ( x ^ - y ^ ) ] - y ) and 2 labeled z^. 20) 'V * =*>' 2 from x v i a + (y .y2»y3'---' n' rri-i' n+i'---'%-i' N Ci y ) n y 3 y = , X 1 , , X r l ' 3'" = (yi.y«y' • ' z + 2 Conti- nue this obtained process after u n t i l £ results. C o r o l l a r y 2.4 t e l l s N - l such o p e r a t i o n s . Since us that £ w i l l be at the i t h step ( i = 1, 2, ..., yj x or N - l ) , we .push x^ downward and ^ ^ upward by the same d i s t a n c e , 1 + 1 1 be an mps of yj . one, x, > m implies that, after each i t e r a t i o n , z. , ( t h e p o s i t i o n where x. , has b e e n pushed l+l l+l r ' right o f y . , so t h a t i+l 6 at the next J always a l e f t w a r d iteration, one. Only N - l , r a t h e r must say the i t h t o ) must be t o the the push of z.,, to y... l+l •'l+l than N, i t e r a t i o n s is are needed be- cause z„ must c o i n c i d e with y„ i f x and y are to have the same mean. N To N show that GCCERA and GMRA are e q u i v a l e n t f o r elementary l o t t e r i e s , we need Lemma 2.3 below: Lemma 2.3: Under completeness, f o r every a, e, 9, p ( e, F Proof: E *4 a-e TW 6 + + L e t q = e/(e+9). F 5 - -e implies that, 9 > 0, p e ( 0 , 1 ] ) , and H e D j , ( 1 ) H "P *4 a-e-9 -k+s+9> 6 >~ + + ( 1 ^ ) H G = " Then, + > pf > P r|Vs-0 + a t r a n s i t i v i t y and SD, GCCERA i £ ( 6 ? a 1 iW + 6 -^ a + + (1-P)H 6 l a+e+9l + (l-p)H > G. Q.E.D. Theorem U5.2 (GRA) : F o r e l e m e n t a r y satisfying Proof: (<=) This ordering > i s straightforward. N m and p r e f e r e n c e completeness, t r a n s i t i v i t y and SD, GCCERA <=> GMRA. (=>) Suppose x = £ . — 6 > ~ i = l N x. l > lotteries y,, w h i c h 2 N £. , — 6 = y. i = l N Jy. Lemma 2.2 t e l l s i by C o r o l l a r y 2.4 i m p l i e s - 50 that £ can be obtained - us that x ~ from x v i a the sequence elementary yj 1 given by _ " I t N i=l y N y N-2 N n-1 _ = E N 1. i=l N n y 1 , N^V i 1 I + i=l ^ J x +E 5 l i E . 1 { following K v C l (Xx i - yy 6 + E ); + i i o N^x, 6 I x n+l ] i J N 1 * 1 i=n+2 N=2 x . < i 6 I x . +S, (x.-y.)J» n n+l i=l l i + n 6 J 1 1 J that v n ••• <,X< <t * * * v l 1 . lF2 y, i i=l y implies l N T-n+2 y r_n-l _ N-2 " T + n _ 1 ^ V>l ± Lemma 2.3 1 Since yj ^ and yj are the lotteries: n-1 *• n X. 1 (2.20). v N-1 _ X. = X,' Q.E.D. The next task is to extend g e n e r a l monetary l o t t e r i e s . Theorem U5.3 (GRA): Theorem U5.2 from elementary lotteries to T h i s can be done v i a CD. Under completeness, transitivity, SD and CD, GCCERA <=> GMRA. Proof: Omitted s i n c e it Theorems U5.1 preferences to be In GMRA <=> that, <=> the U5.3 results on GRA. transitive, Because consistent concavity only r e q u i r e SD (and and do not in addi- depend on 'theory-free', and them w i t h l e t t e r U . <=> of risk GCERA <=> aversion, concavity it of is well u. f o r the more g e n e r a l Fre*chet d i f f e r e n t i a b l e the they with f u n c t i o n a l forms, we r e g a r d them as literature GCCERA are case of Theorem U 7 . 2 . i n d i s t r i b u t i o n f o r Theorem U 5 . 3 ) , preference accordingly label a special complete, t i o n be continuous specific is of local utility functions known t h a t , Machina utility, u(x;F). under E U , (1982a) proves GMRA <=> GCCERA Theorem U5.3 tells us that tal It the e q u i v a l e n c e than b e l i e v e d is between GMRA and GCCERA i s and does not on Fre"chet differentiability. true f o r E U , WU, L I U , FDU, as w e l l as LGU. When more s t r u c t u r e s ger even depend a c t u a l l y more fundamen- implications of theorem on EU i s are imposed on the p r e f e r e n c e global risk aversion For an EU d e c i s i o n von Neumann-Morgenstern u t i l i t y are equivalent: (a) GCCERA; (b) ( C o n c a v i t y ) u(x) (c) GCERA; (d) PLRA. and PLRA. global is utility function, aversion (strong) conditional r i s k aversion to, its are GCCERA; (b) (Concavity) (a) vex i n x . following properties that and an EU d e c i s i o n is consequently referred substitution to. For axiom to definition will the maker i s GCCERA, GMRA, about instance, GRA i f the under weak s u b s t i t u t i o n imply, but will not be he GCERA sense of WU which axiom, a equiva- For a WU d e c i s i o n maker (v,w) with LOSUF C(x;F), the equivalent: (a) Proof: the increasing unconditional counterpart. Theorem WU5 (GRA): following u(x), from E U , we must be s p e c i f i c being weakens the lent following concave; Once we depart risk The maker with a c o n t i n u o u s , function Given Theorem EU5, we may say a concave obtainable. stron- w e l l known. Theorem EU5 (GRA): has are functional, -+ ( b ) : Then, C(x;F) i s Suppose concave there i n x for a l l F; e x i s t s H such that f o r any x^ < x < x^, 2 - 52 there - C(x;H) i s e x i s t s q such strictly that con- C(x ;H)-C(x ;H) ° < C(X3;H)-C(X H) < I; « < X X (2.22) i m p l i e s Define x < 1. X /C(x;H)d6 . > (2.23) (2.24) 1 +(l-q)6 3 2 that 3 F = q6 2 X C(x ;H) < qC(x ;H)+(l-q)C(x ;H). 2 ' X 2~ 1 q < _ 3 l Inequality 2 (2.24) becomes l < /C(x;H)dF, X 2 or /C(x;H)d[F-6 ] > 0. v (2.25) 2 Let G = (l-p)F+pH and G' = (1-p)6 +pH. X 3p W U ( G ) | Since by e x p r e s s i o n (1.9), 2 = /C(x;H)d[H-F] p=1 and d WU(G')| = /C(x;H)d[H-6 ] , dp lp=. x 2 inequality (2.25) i m p l i e s /C(x;H)d[F-6 Since W U ( G ' ) j p = 1 WU((l-p)6 ] = d ?L [WU(G')-WU(G)]| dp = WU(H) = W U ( G ) | p=1 > 0. , we have (2.26) 2 some p s u f f i c i e n t l y close to 1. Since x^ > qx^+(l-q)x^ = JxdF(x) from (2.23), s t o c h a s t i c dominance i m p l i e s WU((1-p)6 ^+pH) > W U ( ( 1 - p ) 6 x (2.26) and (2.27) together tradicting p=1 +pH) < WU((l-p)F+pH) X for x GCCERA. /xdF that +pH)• imply WU((l-p)F+pH) > W U ( ( l - p ) 6 (2.27) /xdp +pH), con- (b) -> ( a ) : Let F , G e D be such that F > z G and d e f i n e F = (l-a)F+aG, vJ a e (0,1). Extend e x p r e s s i o n (1.24) as follows: a = -/C'(x;F )[G(x)-F(x)]dx a (1.24) X = -|C'(x;F )d/_ [G(t)-F(t)]dt 3 a = -JC'(x;F )dT(x) (from ( 2 . 1 3 ) ) a = jT(x)dC'(x;F ) a - /T(x)C"(x;F )dx. Given T(x) (2.28) > 0 f o r a l l x, C(x;F) being concave i n x for a l l F im- a dWU(F ) plies t h a t — ^ — - < 0. By Theorem U 5 . 1 , t h i s y i e l d s GMRA. Since GMRA <=> GCCERA a c c o r d i n g to Theorem U 5 . 3 , we have GCCERA. Q.E.D. Theorems relations PLRA. WU5, U5.3 and for WU: C"(x;F) The f o l l o w i n g Corollary < 0 at example is all 2.3 together F <=> p r o v i d e d to GMRA <=> give the following GCCERA => GCERA => demonstrate that, under WU, PLRA does not imply GCCERA i n g e n e r a l . Example 2 . 1 : (PLRA does not imply GCCERA under WU.) PLRA r e q u i r e s r ( for = - [v 'I( x^ + w(x) ^l ) x) L all x. Pick Consequently, all x. r(x) > 0 = a+bx, = - ^ ^ x (2.29) b > 0, x r ( x (2.30), ; F ) the = e [ 0, = 0 f o r a l l x, ). We have v"(x) the GCCERA c o n d i t i o n reduces term i n the - numerator of 54 - % the =0.) to ( RHS i s = 0. < 0 for i n the case with w'(x) 2 w ' ( x ) v ' ( x ) + w-(x)[v(x)-WU(F)] first 0 0 ^ ^ 0 f o r a l l x, which i m p l i e s w'(x) ( O b v i o u s l y , we are not i n t e r e s t e d Given v"(x) In v(x) 1 always 2 > 3 0 ) nega- tive. For a s t r i c t l y distribution F with w"(x)[v(x)-WU(F)] Similarly, a convex, decreasing WU(F) s u f f i c i e n t l y is sufficiently F with small questions are WU(F) s u f f i c i e n t l y of PLRA imply GCCERA? (2) The interest When w i l l c o n d i t i o n f o r PLRA = _ r ( x ) t at a l l x. C ( to second large here: so term C"(x;F) that > 0. construct C"(x;F) > 0. I (1) Under what conditions will PLRA imply GCERA? > 0 ( 2 . 2 9 ) w(x) x ; F ) w"(x)[v(x)-WU(F)]+2w'(x)v'(x)+w(x)v"(x) JwdF = a l l F , which i m p l i e s , - leading the The c o n d i t i o n for GCCERA i s ' for that a is y ^ +IwHxij v'(x) so construct d e c r e a s i n g w, we can always Hence, PLRA does not imply CCERA i n g e n e r a l . Two we can always positive, f o r a s t r i c t l y concave, distribution w, ! v" , 2w' L —r + V I J > . m after a x t W" w which can be r e s t a t e d as (2.31) Q o m i t t i n g the argument x: I w < v—v r » ~ v' conditions — W V-V i r }» w (2.33) v' /o oo\ (2.32) ' and (2.34) below, n o t i n g w, v' > 0: - | I - The will _ v ^ + — w V" , 2w' — + V W r LHS of Corollary I . > — w — ^ V (2.33) and (2.34) is Clearly, 2.5: „ /o o/N (2.34) . < 0. n the WU A r r o w - P r a t t c o n d i t i o n (2.29) For a WU d e c i s i o n maker (v,w) equivalent: index. PLRA and GCCERA and c o n d i t i o n (2.33) a l i n e a r weight f u n c t i o n w i l l w, the f o l l o w i n g are (a) GCCERA; (2.33) if w r V if i f w" > 0, W" V - V W be e q u i v a l e n t incide. J > or (2.34) co- do. with l i n e a r weight function (b) C(x;F) (c) GCERA; (d) PLRA. Proof: is Omitted. According play concave i n x f o r a l l F ; GMRA, cribe to those who to this GCCERA, EU. corollary, a d e c i s i o n maker's p r e f e r e n c e GCERA and PLRA s i m u l t a n e o u s l y , This recognize suggests the a potential choice restrictiveness of and yet of can does not preference EU but are dissubs- model appalled at to the c o m p l e x i t y of an ' a l l - o u t ' WU. We now t u r n to the second q u e s t i o n : When w i l l PLRA and GCERA be e q u i valent? This A c c o r d i n g to C o r o l l a r y 2 . 5 , is only PLRA w i l l der sufficient however. imply GCERA (but a concave v . they are e q u i v a l e n t when w i s There are not n e c e s s a r i l y A sufficient other conditions GCCERA). c o n d i t i o n for PLRA i s linear. under which For i n s t a n c e , w decreasing. consiRecall that OT r <> - /V /wjx)dF(xS = Mx)dF X) W (x), (1.19) where X / w(t)dF(t) = — £\(t)dF(t) o w F (x) If w is decreasing, F w i l l Lemma 2 . 4 : Suppose the i distribution F, F > F Proof: stochastically weight W (1.20) dominate F function w is w i n the f i r s t decreasing. Then, degree. for any . Omitted. Since case . of strictly (v,w) w i t h v concave E U ) , (v,w) decreasing. will will d i s p l a y GCERA when w i s constant (a be even more i n c l i n e d to d i s p l a y GCERA when w i s In such a c a s e , - 56 a linear v will - suffice f o r GCERA. Given the d i s c u s s i o n C o r o l l a r y 2.6: conditions The If w is holds result above, C o r o l l a r y 2.6 decreasing strictly, is and v i s then PLRA i s s t a t e d without concave and at equivalent proof. least one of the to GCERA. of Theorem WU5 can be extended to the more g e n e r a l LGU: Theorem LGU5 (GRA): For an LGU f u n c t i o n a l V : Dj -»• R with LOSUF £ : J * Dj •* R, the following (a) GCCERA; (b) (Concavity) Proof: are equivalent: C(x;F) is Omitted s i n c e i t is concave i n x for a l l F . s i m i l a r to the proof of Theorem WU5. Comparing Theorems EU5, WU5 and LGU5, larities and function u(x) distinctions. in C(x;F) in both C(x;F) is the function. is the condition First, (b) of Theorem WU5 and non-EU Second, equivalent the we observe von Theorem LGU5, the von both GCERA and PLRA. Third, replaced confirming simiutility by the LOSUF that the LOSUF Neumann-Morgenstern the GCCERA c o n d i t i o n appears s t r o n g e s t form of g l o b a l r i s k a v e r s i o n following Neumann-Morgenstern Theorem EU5 i s of the utility i n a l l theorems because i n the sense that it implies u n l i k e Theorem EU5, the GCERA and PLRA c o n d i - t i o n s are absent i n Theorems WU5 and LGU5 because they are i m p l i e d by, not e q u i v a l e n t In asset the to 'portfolio to, next the GCCERA and c o n c a v i t y section, introduce risk another it we will utilize definition of aversion'. - 57 - but conditions. an agent's global risk demand for aversion, risky called 3 PORTFOLIO CHOICE PROBLEM From a f i n a n c e v i e w p o i n t , in its i m p l i c a t i o n s f o r a s s e t demand. direction approaches under expected depart Definition asset 3.1: with utility from i t . asset-one-risky-asset interest and see how other (gross) rate environment of return r, which refer this functional to a one-safe- section only one safe (gross) r a t e of r e t u r n . The n o t a - are summarized below: r a t e of r e t u r n on the safe z: gross r a t e of r e t u r n on the r i s k y a s s e t ; initial provides set-up. to an a s s e t by i t s gross asset; Q positive x: d o l l a r amount i n v e s t e d i n the r i s k y a s s e t ; y -x: d o l l a r amount i n v e s t e d i n the safe wealth; asset; 6: p r o p o r t i o n of y Q invested i n the r i s k y a s s e t ; 1-8: p r o p o r t i o n of y Q invested i n the safe below is Definition in and one r i s k y a s s e t w i t h r: final preference our i n v e s t i g a t i o n c a l l e d a simple p o r t f o l i o to be used i n t h i s y: the r e s u l t lies world. We w i l l h e r e a f t e r y : i n r i s k aversion We s h a l l present We r e s t r i c t An investment r a t e of r e t u r n z i s tions our u l t i m a t e asset; wealth. 3.2: Problem simple p o r t f o l i o choice To f i n d x* such t h a t , (3.1) an investor's (unconditional) (SPC) problem: f o r every x # x*, F~* > F~, where - 58 - (3.1) y = y r + x(z-r) (3.2) Q y* = y r + x * ( z - r ) (3.3) Q Definition 3.3: portfolio Problem (3.4) choice where investor's conditional F~*+(l-p)H > pFM-(l-p)H, Without a priori that simple given by (3.4) (3.2) and ( 3 . 3 ) , a d i s t r i b u t i o n independent guaranteed an f o r every x * x*, y and y* a r e and H i s is (CSPC) problem: To f i n d x* such t h a t , P below the restrictions p e (0,1], of F . on the o p t i m a l x* w i l l respectively, preference be u n i q u e . It ordering, it is seems d e s i r a b l e not to i m - pose the f o l l o w i n g r e g u l a r i t y : Definition asset 3.4: z, In a simple an i n v e s t o r portfolio with i n i t i a l s e t - u p with wealth y is diversifier / ~ _ \} {F y if his over are s t r i c t l y q u a s i - c o n c a v e r and r i s k y the set of distributions i n x. X o Definition asset 3.5: In a simple portfolio z ,» an i n v e s t o r w i t h i n i t i a l diversifier, {pF y Again, let r + (~_ o us use x the preferences, for wealth y J WU-type r and r i s k y s a i d to be a c o n d i t i o n a l Q his preferences over the set \ + ( l - p ) H } are s t r i c t l y q u a s i - c o n c a v e a simplex indifference are is safe a s s e t of i n x. Z i n t e r p r e t a t i o n of an i n v e s t o r simplex, s e t - u p with c o n d i t i o n a l on p and H , i f distributions point preferences asset s a i d to be an ( u n c o n d i -— o tional) safe of 3-outcome lotteries being a d i v e r s i f i e r . curves are preferences, and - lines are 59 fanning out arbitrary - illustrate Recall that, p a r a l l e l straight nonparallel straight to lines the i n such a for from an nonintersecting EU-type exterior smooth curves 6 y o f o r LGU-type p r e f e r e n c e s . jf are Assume F ~ = (1-p) 6 +p6- and 6 , 6 z z z v z' or t- the three v e r t i c e s of a s i m p l e x , , v r' o where z < r < z and p z + ( l - p ) z > Z r. When x = 0, V(F~) y ral, will this vary. in ensured if x all cross i n x, As x i n c r e a s e s , F ~ w i l l move a l o n g a path and y yZ+p6yZ- . up to (1-p) 6 Q numerous i n d i f f e r e n c e V(F~) w i l l the is In gene- Q curves. If V is strictly e i t h e r be monotone from x = 0 to x = y optimal better-than GMRA EU or WU i n v e s t o r tional, . r 0 A t x = y , the path reaches o path w i l l quasi-concave increase F~ = 6 y y sets x* in and the generically then decrease. simplex are This convex. or Q will be Clearly, a an u n c o n d i t i o n a l , as w e l l as c o n d i - diversifer. Corollary 3.1: A strictly GMRA WU i n v e s t o r is always a conditional diver- sifier. Proof: The CSPC problem f o r a WU i n v e s t o r Max x WU(pFy o r + x f ~-r) + ( 1 ~ p ) H ) The first and second d e r i v a t i v e s and (3.7) below, (v,w) w i t h LOSUF C i s 3 - of WU above w . r . t . x are g i v e n i n in (3.6) respectively: JC(y;G)(z-r)dF(z), (3.6) /C"(y;G)(z-r) dF(z), (3.7) 2 where 5 ( - ) G = pF^*f(l-p)H y for all quasi-concave G e Dj, and y = y r + x ( z - r ) . Q (3.7) is always With C(y;G) s t r i c t l y negative, i.e. WU i s concave strictly i n x. Q.E.D. On such the contrary, a property. an LGU f u n c t i o n a l For example, Dekel - 60 V(F) does (1984) - showed not generally that, possess under FDU, the concavity of l o c a l u t i l i t y preference f u n c t i o n a l V ( F ) are j o i n t l y ences over a s s e t s vestor f u n c t i o n s u(x;F) and the q u a s i - c o n c a v i t y being sufficient to be q u a s i - c o n c a v e . a diversifer means some degree of a r b i t r a r i n e s s i n the curves folio d e c i s i o n s might y i e l d u n d e s i r a b l e We define the x* that prefer- For LGU, the requirement of an i n - indifference choice i s removed that f o r the demand of the so as to r u l e out cases where simple solves multiple an i n v e s t o r ' s port- solutions. SPC problem (3.1) or CSPC problem (3.4) as h i s demands f o r r i s k y asset z: Definition 3.6: Suppose x* s o l v e s i n v e s t o r with i n i t i a l the SPC problem wealth y . money r i s k y - a s s e t demand at y Q (3.1) u n i q u e l y f o r an Then, x* i s c a l l e d h i s ( u n c o n d i t i o n a l ) and 8* = x*/y Q i s c a l l e d h i s (uncondi- t i o n a l ) p r o p o r t i o n a l r i s k y - a s s e t demand at y . Definition investor 3.7: Suppose x* s o l v e s with initial wealth money r i s k y - a s s e t demand at y Q the CSPC problem y . o (3.4) u n i q u e l y f o r an T h e n , x* i s c a l l e d h i s c o n d i t i o n a l and 8* = x * / y Q i s called his conditional p r o p o r t i o n a l r i s k y - a s s e t demand a t y . When i t i s unambiguous, we may omit above terms. the Note that we do not r u l e out s h o r t s a l e s CSPC problems. assumed 'money' and ' p r o p o r t i o n a l ' i n the throughout This this however w i l l essay that i n both the SPC and not be an i s s u e here because i t i s E ( z ) > r , which implies that x* i s always n o n n e g a t i v e . Theorem U6 (Nonnegative C o n d i t i o n a l R i s k y - A s s e t the CSPC problem Demand): Suppose x* s o l v e s (3.4) f o r any i n v e s t o r whose p r e f e r e n c e s t r a n s i t i v e and e x h i b i t SD and GCCERA. Then, (1) x* (8*) > 0 i f E ( z ) > r ; moreover, x* (8*) = 0 (2) x* > 0 o n l y i f E ( z ) > r . - 61 a r e complete, - i f E(z) = r; Proof: (1) Suppose E ( z ) E(y*) > r but x* < 0. = y r+x*[E(z)-r] < y r. Q SD i m p l i e s t h a t , Then, Q f o r any p e [0,1) a-p)S ~*)+PH < (1-P)6 and H e D , +pH. E( GCCERA i m p l i e s (^^(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 ~ * + p H . o T h i s c o n t r a d i c t s the o p t i m a l i t y of When E ( z ) = r, E(y) x*. = y r+x[E(z)-r] Q = y r f o r a l l x. Q GCCERA i m - plies (l-p^^+pH = d-p)6 y r +pH > ( l - p ) F M - p H , i m p l y i n g x* = 0. (2) Suppose x* > 0 but E ( z ) < r . of (1) a b o v e . ) Then, E(y*) t r a n s i t i v i t y imply that (l-p)6 y r (We need not c o n s i d e r E ( z ) = r i n l i g h t = y r+x*[E(z)-r] Q < y r. Q GCCERA, SD and +pH > ( l - p ) 6 ~ ^ + p H > ( l - p ) F ~ + p H , E ( ) A c o n t r a d i c t i n g o p t i m a l i t y of x* > 0. Q.E.D. The r e s u l t s of Theorem U6 can be o u t l i n e d as a. E(z) > r -> x* > 0; b. E(z) = r -> x* = 0; c. x* > 0 •*• E(z) > r . - 62 - below: Note that x* > 0 i s sufficient implies which i n t u r n i m p l i e s x* > 0. can E(z) > r, be e s t a b l i s h e d but not necessary f o r E ( z ) > r s i n c e x* > 0 The e q u i v a l e n c e under EU and WU. For an EU maximizer, the CSPC problem becomes: Maximize x E U [ p F H - ( l - p ) H ] = p / u ( y ) d F ~ ( y ) + ( l - p ) Ju(s)dH(s) s.t. y = y^r + x ( z - r ) . The however y (3.8) y CSPC problem f o r a WU maximizer w i t h value f u n c t i o n v and weight function w i s : Maximize x WU[pF~Kl-p)H] where y = y r + x(z-r). The two (3.5) y Q optimization conditions f o r (3.8) and (3.5) lead to the following theorems: Theorem EU6 ( P o s i t i v e the Conditional Risky-Asset CSPC problem (3.8) for an EU i n v e s t o r concave u t i l i t y f u n c t i o n u ( y ) . x* (8*) > 0 if Demand): Suppose with an i n c r e a s i n g , x* solves strictly Then and o n l y i f E ( z ) > r . P r o o f : Omitted s i n c e i t is a special case of Theorem WU6. Also, see Arrow (1971). Theorem WU6 ( 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): the CSPC problem (3.5) f o r a WU i n v e s t o r concave LOSUF C ( y ; F ) . Then, x* (8*) Proof: fine In l i g h t > 0 if and only i f (v,w) Suppose with i n c r e a s i n g , x* strictly E(z) > r . of Theorem U6, we need to prove only the s u f f i c i e n c y . G = pF~f(l-p)H. solves The FOC and SOC f o r x* to s o l v e (3.5) - 63 - De- are given by by (3.9) and (3.10) below, respectively: FOC: / C ' ( y * ; G ) ( z - r ) d F ( z ) = 0; SOC: JC"(y*;G)(z-r) dF(z) where C(y;G) - w(y)[v(y)-WU(G)]//wdG and y* 2 < 0. (3.10) = y r + x*(z-r). Q Suppose E ( z ) > r but x* = 0. < 0.) (3.9) (Theorem U6 r u l e s out the p o s s i b i l i t y of x* Then, /C'(y*;G)(z-r)dF(z) c o n t r a d i c t i n g FOC ( 3 . 9 ) = C'(y r;G)[E(z)-r] o > 0, and i m p l y i n g x* > 0. Q.E.D. Theorem WU6 t e l l s us t h a t , l i k e his expected u t i l i t y c o u n t e r p a r t Theorem E U 6 ) , a GCCERA WU maximizer w i l l only if the expected return on the r e t u r n on the r i s k f r e e a s s e t . demand w i l l of fact, the l a t t e r The of imply p o s i t i v e results GCCERA. If 3.8: asset i n the is Obviously, positive of As a matter GCCERA). i n Theorems U6, EU6 and WU6 are based on the is risk risky-asset In a simple seeking, then i t demand when E ( z ) is > r. This suggests yet ano- aversion: portfolio s e t - u p with safe a s s e t (unconditional) portfolio Q assumption q u i t e n a t u r a l f o r him display demand a t y sure risky-asset demand. the p r e f e r e n c e of an i n v e s t o r w i t h i n i t i a l is than the conditional asset z, risky-asset r i s k y a s s e t i f and greater unconditional risky-asset t h e r c h a r a c t e r i z a t i o n of r i s k Definition risky r e q u i r e s o n l y GCERA ( i n s t e a d an agent to have p o s i t i v e invest (cf. is risk positive 64 it - Q (PRA) a t only i f E(z) > r . s a i d to d i s p l a y g l o b a l PRA (GPRA) i f - aversion wealth y r and r i s k y is y said Q if to his His preference d i s p l a y s PRA at a l l y . Definition asset 3.9: z, the p r e f e r e n c e display tional conditional risky-asset ference all is said true. sense: 0.5, of an i n v e s t o r demand at y to s e t - u p with display is Q positive global safe a s s e t r and r i s k y with i n i t i a l wealth y p o r t f o l i o r i s k aversion stronger GCPRA i s If (CPRA) at y only i f CPRA (GCPRA) then he i s According conditional tional) therefore if Q if Q is said his E(z) > r . it to condi- His p r e - displays CPRA at GPRA but the converse is to (unconditional) risk light Theorem U6, in 3.2: the Under it dering As imply it turns satisfying conditional following Lemma 3 . 1 : (3.8), GCPRA. not risky-asset but, risky is if asset, clear that preference completeness, GCPRA f o r a c o n d i t i o n a l GCPRA 3.9 averse, GCPRA no matter whether h i s theories? CPRA when p = 1, GPRA but not Definition amount Does is than GPRA i n the following but not so when p = N a t u r a l l y we expect to d i s p l a y i n g GPRA than those d i s p l a y i n g GCPRA. positive Corollary more r e s t r i c t i v e by d e f i n i t i o n porfolio of than GPRA as GCPRA i m p l i e s f o r any y , an I n v e s t o r f i n d more people the portfolio y . GCPRA i s not In a simple all demand a it investors are GCPRA or with positive conditional GPRA agent (uncondiinvests a must be true that E ( z ) > r . any GCCERA i n v e s t o r functional transitivity is must In also be E U , WU, FDU or LGU. and SD, GCCERA implies diversifer. GCCERA i n out, or GCPRA i m p l i e s completeness, diversifier general under GCCERA f o r transitivity, assumption. particular To any p r e f e r e n c e stochastic show preference dominance, this, we need orand the lemma: Suppose x* s o l v e s u n i q u e l y - 65 the - CSPC problem ( 3 . 4 ) for a condi- t i o n a l d i v e r s i f e r whose p r e f e r e n c e s w i t h SD and e x h i b i t Proof: -x* Suppose (> 0) GCPRA. x* < 0. is are complete, Then, x* = 0 i f (The d i f i n i t i o n of transitive, E(z) = r . GCPRA r u l e s optimal for r i s k y asset z' consistent = 2r-z, out x* > 0.) Then, c o n t r a d i c t i n g the defi- n i t i o n of GCPRA. Q.E.D. Theorem U5.4 equivalent Proof: (GRA): Given C o r o l l a r y exist where y is E(y)/y 2.2, p e (0,1], the r.v. and SD, GCCERA is diversifer. to prove GCPRA > GCCERA. Suppose E associated with F . For a g i v e n y , c o n s t r u c t condi- demand i n the CSPC problem with the parameters y , r, Then pF i m p l i e s x* = 0, Note that E ( z ) = r . r = L e t x* be the Q above. suffices transitivity F , H £ Dj such that p F M - ( l - p ) H >- p 6 ^ ~ ^ + ( l - p ) H , y Lemma 3.1 it and z = i - ( y - y r ) + r . tional risky-asset given completeness, to GCPRA f o r a c o n d i t i o n a l there z Under ~ + ( l - p ) H > F ~ f ( l - p ) H >- po +(l-p)H. P y o giving rise y o to a c o n t r a d i c t i o n . Q.E.D. - 66 - 4 COMPARATIVE RISK AVERSION In ber of the proceeding ways. sections, The q u e s t i o n to we c h a r a c t e r i z e d r i s k explore d e c i s i o n maker being more r i s k a v e r s e ral implications? 4.1 Definitions Since concavity GRA, it (a) of is CCE and C E , (b) n a t u r a l to t h i n k of (c) extended averse accept we say (CE) (b) What are i t s demands, have been used to We w i l l c o n s i d e r one behavio- and (d) the characterize for characte- them one by one. Certainty Equivalent risk to i n a num- the meaning of them as p r o m i s i n g c a n d i d a t e s Both CCE and CE are concept ly What i s risky-asset functions r i z i n g comparative r i s k a v e r s i o n (CRA). (a) is: than another? mps, appropriate u t i l i t y next aversion to a comparative of a s i n g l e risk than agent B i n the a lower that CCE (CE) f o r agent A i s aversion sense of therefore context. If can be agent A i s easimore CCE ( C E ) , we w i l l expect agent A any d i s t r i b u t i o n than agent B. more GCCERA (GCERA) of any d i s t r i b u t i o n i s value, than agent B i f Formally, agent A ' s CCE s m a l l e r than agent B ' s . P r e f e r e n c e Compensated Spread Mean preserving comparative risk spreads aversion. do not work Recognizing quite this, well in characterizing Diamond and S t i g l i t z proposed a 'mean u t i l i t y p r e s e r v i n g s p r e a d ' n o t i o n (they c a l l e d i t - 67 - (1974) a 'mean utility tion preserving preserves Increase 'mean u t i l i t y ' , rative risk ences, more r e s t r i c t i v e is aversion s i m i l a r to that Definition of 4.1: for it as (ii) preferences. definitions name s a y s , to characterize For more are needed. this general The f o l l o w i n g no- compaprefer- definition (1982a): Distribution G is said G s i n g l e - c r o s s e s F from the to be a simple compensated spread left, and F ~ G. Compared with Definition trictive (ii), as it however, d e c i s i o n maker, sated spread, compensated Diamond and 4.5 below), only is expected u t i l i t y allows more ones. there i.e., Stiglitz' condition (i) in not Depending on the least three utility which once. more res- Condition preferences to preferences subscribed to by different the spread, and LGU s a i d to be a simple mean u t i l i t y preser- spread, cases of only compen- Distribution G is WU compensated d i s t r i b u t i o n F to an expected u t i l i t y G s i n g l e - c r o s s e s F from the left, deci- function u i f and EU(F) = EU(G). D e f i n i t i o n 4.3: tion is restricting s i o n maker with von Neumann-Morgenstern u t i l i t y (ii) cross 4.1 spread simple EU compensated v i n g spread ( s i m p l e mups) of (i) preserving in Definition distributions general are at mean spread. D e f i n i t i o n 4.2: w, (simple (a) the can only be used EU-type i n Machina Since, d i s t r i b u t i o n F to a d e c i s i o n maker i f (i) (cf. in r i s k ' ) . G is For a WU maximizer with v a l u e said to be a simple weighted wups) of F i f G s i n g l e - c r o s s e s F from the left; - 68 and - f u n c t i o n v and weight utility preserving funcspread (b) WU(G) = WU(F). Definition 4.4: Distribution G is said to be a simple LGU p r e s e r v i n g spread of d i s t r i b u t i o n F to an LGU d e c i s i o n maker V i f (i) G s i n g l e - c r o s s e s F from the l e f t , (ii) and V(F) = V ( G ) . Since EU i s linear for second-degree via Definition permissible in distribution, stochastic 4.5 the squeezed mean dominance can be g e n e r a l i z e d below. This will significantly interpretation to mean u t i l i t y increase the set of distributions. D e f i n i t i o n 4.5: Distribution G is a mean u t i l i t y p r e s e r v i n g spread (mups) of d i s t r i b u t i o n F to an EU d e c i s i o n maker u i f y J_ u(x)dF(x)+u(y)[l-F(y)] > J_ u(x)dG(x)+u(y)[l-G(x)] y for a l l y, (4.1) and u(x)dF(x) Condition utility. F over ( - 0 O = (4.2) the , y ] be not l e s s than that that 'squeezed' of G f o r a l l a risk factor is (Diamond and S t i g l i t z ) : function serving increase u(x), an i n c r e a s e in risk u'(x)F (x,a)dx a a. where a s u b s c r i p t to which a p a r t i a l d e r i v a t i v e = /JL the F and G y i e l d equal expected in expected utility of y. a sequence of mups of F , Diamond and S t i g l i t z as D e f i n i t i o n 4 . 5 ' , utility (y) distributions requires D e f i n i t i o n 4.5' T that d i s t r i b u t i o n with restated respect says (4.2) C o n d i t i o n (4.1) To c o n s i d e r ized j_2 u ( x ) d G ( x ) . Definition indicates 4.5 parametercan then be the v a r i a b l e with taken. Given a d i s t r i b u t i o n F ( x , a ) a represents a mean u t i l i t y and a pre- if = /_ y F (x,a)du(x) a - 69 - > 0 for a l l y, (4.3) and T ( » ) = j_l u ' ( x ) F ( x , a ) d x a To gain some a)F+aG, where F a is to insight a e [0,1] = j_l F ( x , a ) d u ( x ) = 0. a into Definition 4.5', and G i s (4.4) t t define F a mups of F as d e f i n e d by D e f i n i t i o n a mixture of F and G with component of G i n c r e a s i n g as 1. In other words, { F : a e [ 0 , l ] } represents a a sequence going from F towards G. Since the same expected u t i l i t y i s F [(l-a)F+aG] to G and F (x,oc) = ^ a equivalent to c o n d i t i o n s F(-°°)-G(-») (4.1) = G-F, conditions and ( 4 . 2 ) , respectively, expected risk increase if agent spread, that averse 4.5. a goes from 0 of mups of F , preserved from (4.3) and (4.4) are noting F ( » ) - G ( » ) = = 0. Suppose that agent A i s more r i s k averse compensated = F(x,oc) = ( 1 - and that G will agent G is a compensated Formally, A always spread of F to A . be p r e f e r r e d to F from B's viewpoint should i n g e n e r a l in risk. than agent B i n the sense of prefers demand l e s s we say that F to any of agent agent because compensation It is a less for a given A i s more MRA than agent B B's preference compensated spreads of F . (c) R i s k y - A s s e t Demand In a simple p o r t f o l i o s e t - u p , his conditional only i f than the with the the expected risk-free respect seems (unconditional) to reasonable risky asset. we d e f i n e d an agent risky-asset to be CPRA (PRA) i f demand i s strictly r a t e of r e t u r n on the r i s k y a s s e t i s rate of return. z and r . If to the expect Formally, strictly greater Suppose both agents A and B are CPRA they have i d e n t i c a l i n i t i a l more r i s k we say that - positive 70 averse agent - agent to wealth, then it demand l e s s of A i s more GCPRA (GPRA) than agent B if for any r and z s u c h that E ( z ) > r agent A ' s demand f o r z always l e s s than agent B ' s . (d) C o n c a v i t y o f r e l e v a n t u t i l i t y Suppose more s t r u c t u r e s that a GCCREA LGU5). agent must functions are imposed on a p r e f e r e n c e have a Agent A being more r i s k A's u t i l i t y function, Definition least 4.6: as g if as Lemma 4.1 identifiable, (Pratt): exists f(x) utility is an function another increasing (cf. B suggests 'more concave' than) f u n c t i o n a l V so function than agent continuous (more concave there f u n c t i o n h such that concave averse An i n c r e a s i n g , concave function if is f that agent than agent B ' s . is said increasing, concave Theorem to be at continuous (strictly concave) = h(g(x)). Suppose f and g are two concave, increasing functions. Then, -f'7f if > ( » and o n l y i f -g'Vg' f is (4.5) at l e a s t as concave as (more concave than) In words, D e f i n i t i o n 4.6 means that f can be obtained h. by 'concavifying' if f i s more concave g. than g, g v i a an i n c r e a s i n g concave then function F o r EU m a x i m i z e r s u^ and u ^ , u^ being more concave than u ^ , by Lemma 4.1, implies that - u " / u ! A this means that A since B the more r i s k A r r o w - P r a t t index w i l l Beyond > -u"/u'. EU, the Since - u " / u ' is the A r r o w - P r a t t index, B averse an EU i n d i v i d u a l i s , the g r e a t e r his be. 'concavity the LOSUF C(x;F) serves as index' will n a t u r a l l y be ~ C " ( x ; F ) / C ( x ; F ) the von Neumann-Morgenstern u t i l i t y - l i k e function. - 71 - 4.2 Characterizations Now that we have clarified the meaning of one agent GCCERA, more GMRA, or more GCPRA than another agent, this section turns more We to e s t a b l i s h the r e l a t i o n s , GMRA than first then extend Definition B regardless show that of this the p r e f e r e n c e is true i t to g e n e r a l monetary 4.7: G i s a simple any, among an elementary theory f o r elementary them. they i f A is also subscribe to. monetary compensated a, e, p , 9^, 9 l o t t e r y H such that 2 (e, spread 9^, 9 lotteries he i s i n d i f f e r e n t of F to a > 0, p e (0,1]) 2 between = ( l - p ) H + p { i 6 a _ e+ i 6 a + e } F As i t lotteries. elementary d e c i s i o n maker i f there e x i s t and if more the remaining task i n o u t , agent A i s more GCCERA than agent B i f and only shall and is being (4.6) and G - (l-p)H + p{K-e-91 In Definition crosses F only 4.7, the once G are elementary (although lotteries comes. In D e f i n i t i o n strictly risk desirable shift 9 seeking 2 averse than F . Definition word 9 >' 4.7, i f 4 'simple' they is used to indicate involve a finite G will 2 spread an mps of a d i s t r i b u t i o n F i s than the l e f t - t a i l of x i f number of shift F and out- be an mps o f F . F o r In o r d e r to make a spread as a t t r a c t i v e , the o p p o s i t e that G might be an i n t e r v a l ) . 9^ = 9 , makers, 7 < ' > 2 the c r o s s i n g 0^. less the r i g h t - t a i l For a s t r i c t l y risk i s t r u e ( i . e . 9^ > 9 ) . 2 4.8: F o r e l e m e n t a r y compensated + e + because decision must be g r e a t e r individual, k + lotteries there - exists 72 - x, £ e J N , £ i s an elementary a nonnegative compensating vector t , TV,,..., 1^_-Il ) such that rj = ( n .1 N-1 _ where x x - ( ^» 2»••*» N^ z = (y » L x 1 X x + T x i » 3»---» 2 n1 x, = (?i x, n 1 N ) Vi>VVi'Vi>-'V + = (yi.---.y - .y .« Z x 1 1 n .i \» t r i x = (y ,...,y ,...,y _ ,x + 1 n N 1 1 N x n f2*-- N ) V l = ^ ) ( , . . . , y , . . . . y ^ ,y ) n N 1 Note that yj ^ and yj are r e s p e c t i v e l y G _ ^ and n-1 T = { E i = l i G _ N-2 r n-1 1 . E " i n i = l N=2 V n J G n is + E } + { + i + y i spread of x, compensated must true rightwards served. the that to, n n+l distance x^ if £ is G^ ~ G ^ _ ^ . an e l e m e n t a r y Defi- compensated ele- > y^. First such that push x^ l e f t w a r d s it to y^ and push the d e c i s i o n maker's p r e f e r e n c e i s pre- d e n o t e the d i s t a n c e of the l e f t w a r d push at step i by of the simultaneous r i g h t w a r d push by n^. x x^-y^> 0, and to that, . n spreads i n the f o l l o w i n g manner: S t a r t i n g with x, say, L e t us leftwards us z 1 t h e n £ can be o b t a i n e d from x v i a a sequence of simple mentary be tells A 6 n 4.8 t h e r e f o r e = ) * n+l , '* 6 a s i m p l e elementary compensated spread of G _^ i f nition and N=2 X, ¥ T x +TI + K i n n-1 N 1. i .2,1. 1 i=n+2 N ^ x , ' N^y 2 x ^+n 6 8 - g i v e n below: n G 4 > l~yi , ^ t * i e d e c i s i o n maker i s and x^ r i g h t w a r d s - 73 to - z^. This GMRA. time, C l e a r l y , X^= Next, push \^ = 2~y2 z = (x +T i ) - y 2 1 = n +(x -y ). 2 1 2 d e c i s i o n maker i s 1 1 1 = \_i ~y > +x ± ± \ ± Vi'Wi'Vi'Vz X = \ > \ 0 > i f t h e GMRA, and yj " = ( 7 i yj In g e n e r a l , 2 X (yi»"-.y - .Wr n' n+l n + T 1 1 V X x n x + T ) + = (y , • • • . y ^ - L • y » + l V i » n + 2 l X n' n 2»--" N X n N> N-l for Vl n = 2, = y N In _ X 3, > N N-l. Since x £ +n must be £ , we have jj jj_j_ = y^» o r °' light of the definition of GMRA, the following corollaries are obvious: Corollary 4.1: £ TJ, is For a r i s k n e u t r a l ( i n the sense of mps) d e c i s i o n maker, i f an e l e m e n t a r y compensated spread of x v i a the compensating v e c t o r then = Corollary GMRA 4.2: Suppose decision consistent f ^ ^ ( X i - y j ^ ) o £ is r all 1 < k < N-l. an elementary compensated spread of x for a m a k e r whose p r e f e r e n c e with SD. Then, the is complete, transitive and compensating v e c t o r TJ s a t i s f i e s the following: v > ^ ^ ( X j - y . ^ for all 1 < k < N-l. x P r o o f : We prove by i n d u c t i o n . Suppose n^ < ^ - y ^ . Then SD and GMRA imply that X, 1 = (y ,x +n ,x ,...,x ) 1 2 L 3 -< ( y , x + ( x - y ) , x N 1 2 1 1 3 —< ( y ^ + ( x ^ — y ^ ) , x , x , . . . , x ^ ) = ( x ^ , x , x , . . . , x ^ ) 2 contradicting x ~ y / . 3 2 3 T h e r e f o r e , i t must be t r u e that - 74 - x ) N = x, x > -^-y^. k Next, suppose, ^ ( x ^ y ^ . k + 1 x k < N , n^ > - ( -£ y£) but \ + i ^ Then, - ( X, f o r some a. f v y + X ~ (y , • • • > k ' k + l 1 -< x yk'\+l>\+2 Vl' k+3 - (yi'-*-»yk , y x X x X k+l' k+2' k+3''*' + x ^ ) N ) » N x x k+l \+l' k+2' k+3'"*' N^ ^ b y G M R A a n d S D by S D ^ k+1 + x -< (yi>->->y >y i h i(x -y )>x >\ 3>--'> ^ k k+ = -< ( y i . - - - » y » y k = giving tion, ( y rise l y k , X k + l + , \ ± + k + X ± + i \ ( k + 2 k+2 x k + , X ry k + i)» x k + 3 ' * " to a c o n t r a d i c t i o n . < + x k + , X 2' k+3 N ) = b y S D ) u c ~ Z x Thus, ( V ) y > ^L=l^ i~ i^* B y n l ^ we have proved that > ^jti^i-yi) f o r a l l 1 < k < N. Q.E.D. Lemma 4 . 2 ; L e t >^ and >g be the r e s p e c t i v e A and B which s a t i s f y GCCERA than B , t h e n , completeness, p r e f e r e n c e o r d e r i n g s of agents t r a n s i t i v i t y and SD. f o r any elementary l o t t e r i e s I f A i s more F and G such t h a t G i s a simple elementary compensated spread of F f o r A , G >g F . Proof: Since, definition, e (0,1]) F f o r A , G i s a simple elementary there must e x i s t spread of F , by a , e, 9^, 9 , p ( e , 9^, 9 > 0, p 2 and an elementary l o t t e r y H such that F ~ ^ G and E(i- p ) H + p{l 6 _ + l a q^, q 2 6 £ G =(l-p)H + p{jVc-9 Let constants compensated + 1 a + £ }, 6 y a+s+9 ^ e [0,1] be such t h a t 2 2 q ~ That B i s plA-e-^ (i-P) + A q i + q q i ? "ArX + + t 1 " l e s s GCCERA than A i m p l i e s G > B Q > A ? + e + e l E 2 Q P;">^ F . B Q.E.D. Given F defined downward s h i f t Theorem (CRA): orderings (a) (4.6), 9^, B w i l l U7.1 following by > A elementary satisfying B c o n d i t i o n s are refer to lotteries If l o t t e r y F , CCE (F) E . ^ , 4° i = l N x, Proof: can (a) be 1 (b) -> ( a ) : lotteries l This F and pair of v is H, preference t r a n s i t i v i t y and SD, the < CCE (F), and B A respectively. an elementary compensated spread of x = F at l e a s t as p r e f e r a b l e as y to B . v an elementary compensated spread of x to A, £ from x v i a a sequence of simple elementary spreads yj g i v e n by ( 4 . 8 ) . - O v a CCEs of F ( c o n d i t i o n a l upon any p e [0,1] is y Since £ i s obtained an i d e n t i c a l and to A , then x i s ~ (b): for where C C E ( F ) A the - ^5° £ = G = and completeness, an elementary l o t t e r y H) f o r A and B, (b) that, equivalent: F o r any e l e m e n t a r y CCE_(F) implies r e q u i r e a lower compensation 0£ than A w i l l . For and > Lemma 4.2 x compensated A c c o r d i n g to Lemma 4 . 2 , n N-l _ v straightforward recognizing that, pF+(l-p)H i s an elementary 1 'W -""' f o r any elementary compensated spread of ... P E D - 76 - Given (b), Lemma 4 . 2 , lottery We are Theorem U7.1 is obvious. Note that, in condition G need not be a simple compensated spread of F . now ready to further extend Theorem U7.1 to g e n e r a l monetary lotteries. Theorem U7.2 orderings (CRA): >. The f o l l o w i n g and > „ which are complete, t r i b u t i o n and c o n s i s t e n t (a) (GCCERA) and (b) F o r any CCE (F) r e f e r B (GMRA) If G is (a) Consider -»> ( b ) : n {i/2 : for a pair transitive, of preference continuous in dis- with SD: d i s t r i b u t i o n F , CCE (F) < CCE (F), A B to the CCEs of F f o r A and B, a simple compensated t i o n F to A , then G i s Proof: are e q u i v a l e n t Suppose at least G is a n i = 1, 2 - l }. n compensated L e t p ° = F(0) sup {X|F(X)=P°} p° n < i/2 n inf {X|F(X)=°} spread of and q ° = G ( 0 ) . F to A . Define 0 p° is n < p +(l/2 ) p°+(l/2 ) if n _ x„ = distribu- < p° n n A respectively. spread of an a r b i t r a r y simple 0 < i/2 sup { x | F ( x ) = i / 2 } CCE (F) as p r e f e r a b l e as F to B . inf {x|F(x)=i/2 } n where < i/2 n < 1, not i n I n = {i/2 |i=l n 2 -l} otherwise P and n infty|G(y)=i/2 } 0 < i/2 sup { y J G ( y ) = q ° } p° n sup { y | G ( y ) = i / 2 } f 0 i n f {y F ( y ) = q ° } n < i/2 < q° n n q°+(l/2 ) if n q°+(l/2 ) < i/2 n q ° i s not i n I otherwise. - < 77 - < 1, - n {i/2 Ii=l,...,2 -l} n N — 91 N — 9—1 F = 2 6n, +TS7 2 ~ 6 n and G = 2 6 n , +jJL 2~ 6 n, n x. + e i=l x.+e n y«+9 i=l y. + 6 O n i n 0 n •'i n 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 ° n Construct n n n 1 a 0 J such that tion to F and G, r e s p e c t i v e l y . G single n ° c r o s s e s F from the n To prove G > left. F , suppose B Furthermore, for 3 all k > K. A g a i n , by CD, there " * » » • ' » all m > M. Pick D J G. Since e x i s t s a K > 0 such that lim^^F^ F. >-_ G f o r k B e x i s t s an M > 0 such that F, >G for k B m N = maxlK.M}. Then, F >- n (b) large, By Theorem U 7 . 1 , G > F . ' n ~B n the c o n t r a r y that F >- = F and l i m G = G, by CD, there n-*» n ' ' rise n sufficiently G B f o r a l l n > N, g i v i n g n to a c o n t r a d i c t i o n . ->• ( a ) : pF+(l-p)H i s a simple compensated spread of + po^g (pN (^-~P) H A to A . C o n d i t i o n (b) pF+(l-p)H >g pF+(l-p)H ~ implies P 6 C C E ( F ) P 6 C C E that +(l-p)H. Since by B ^ ( F ) +(l-p)H, transitivity, p 5 CCE*(F) SD i m p l i e s + ( 1 - p ) H CCE (F) 6 P CCE(F) + ( 1 _ P ) H - > CCE (F). B A Q.E.D. We have if A is proved that agent A is more GCCERA than agent a l s o more GMRA than B r e g a r d l e s s functionals tribution, as long as they are of complete, and c o n s i s t e n t w i t h s t o c h a s t i c We next consider 78 transitive, their and o n l y preference continuous in dis- dominance. comparative GCPRA. - the forms of B if It - appears that an i n v e s t o r who is more GCCERA than what u t i l i t y (CRA): Suppose transitive, Then, A i s Proof: investor will theory t h e i r p r e f e r e n c e s Theorem U7.3 plete, another the continuous also subscribe preferences in be more GCPRA no to. of agents distribution and A and B are consistent more GCPRA than B i f A i s more GCCERA than B . In a s i m p l e portfolio x^+Ax are the r e s p e c t i v e initial P F r set-up w i t h E(z) wealth y . risky-asset > r, suppose x _•_/ . A w ~ N+(1-P)H y r+(x.+Ax)(z-r) o A v follows that there exists N >_ ~B v ~ J Suppose r r spread Ax < 0. of v Theorem U 7 . 2 , PF v that ^ ,~ »+(l-p)H. y r+x,(z-r)+9 ' o A N j N l r v will v to B . v be a compensated * Since A i s more GCCERA than A SD and t r a n s i t i v i t y imply t h a t ./ . » w~ N+(1"P)H y r+(x.+Ax)(z-r) o A v J >. ~A v pF , ,~ N. +(1-P)H. y r+x.(z-r)+9 ' o A Q v >-. pF A y J This contradicts r pF , ,~ .+(l-p)H y r+x.(z-r)+9 o A s that L r w identi- J pF B and x = B pF _ ,~ .+(l-p)H. y r+x.(z-r) o A pF . . ~ +(l-p)H y r+(x +Ax)(z-r) v o B, Then v 9 > 0 such PF j_, .A ~ +(l-p)H * y r+(x.+Ax)(z-r) *' o A w demands of A and B who have The o p t i m a l i t y of x^ i m p l i e s Q J It com- with SD. A cal matter . ,~ +(l-p)H. rfx.(z-r) A N o v the o p t i m a l i t y of x. to A . A v Hence, Ax > 0 and x „ > x „ . B A Q.E.D. Theorems EU7, WU7 and LGU7 below t e l l B also us that A being more GCPRA than i m p l i e s A being more GCCERA than B under E U , WU and LGU. Theorem EU7 (CRA): The f o l l o w i n g are e q u i v a l e n t for a pair i n c r e a s i n g von Neumann-Morgenstern u t i l i t y f u n c t i o n s (a) (GCCERA) F o r any d i s t r i b u t i o n F , CCE (F) - 79 - of continuous, u ^ and u^: < CCE ( F ) , where CCE (F) and (b) CCE (F) refer (GCPRA) In asset z, wealth a simple portfolio where E ( z ) and x^ and a s s e t demands. (c) ( C o n c a v i t y ) (d) to the CCEs of F f o r u B u > r, x^ are any CE (F) refer D to the CEs (e) (GPRA) asset In z, a r and u* have i d e n t i c a l risky initial risky- u„, r e s p e c t i v e l y . simple portfolio > r, Xg are x^ and Then x. a set-up with suppose u^ and safe asset r and u^ have i d e n t i c a l risky initial t h e i r r e s p e c t i v e u n c o n d i t i o n a l money r i s k y < x_. B f o l l o w i n g lemma i s needed to prove Theorem Lemma 4.3: then asset Omitted s i n c e t h i s i s well-known. The (ii) safe F, CE.(F) < CE„(F), where CE.(F) and A B A of F f o r u. and A (i) u and A. a s s e t demands. Proof: with t h e i r r e s p e c t i v e c o n d i t i o n a l money distribution where E ( z ) wealth and suppose u^, r e s p e c t i v e l y . Then x, < x „ . A B i s at l e a s t as concave as u . (GCERA) F o r D set-up and A If G s i n g l e - c r o s s e s F at x* f ( x ) and g(x) are two from the l e f t ; g(x), J[G-F]f'(x)dx f'(x*) J[G-F]g'(x)dx. g'(x*) Given such that x f f(x) n (x ) 2 > at least < x„, we and i n c r e a s i n g f u n c t i o n s , and concave i n x as Proof: as concave have g '(x ) 2 straight algebra yields - the 80 following: - f ( x ) i s at l e a s t i n x as g ( x ) , f o r any g Applying WU7. x^, x^ as £ R [ G - F ] f (x)dx £2 = f* < * * > £ ! [G-F] f^gy - *'(**) UfjG-F] > f»<x*) {f*[G-F] i;_co f fx*) L dx f g j y dx + j £ [ G - F ] f ^ g j dx} g'(X*) dx + / ^ [ G - F ] i ^ O X * g'(x*) V J L dx} ' 00 1+ = frgrf- JI. [G-F]g'(x)dx. Q.E.D. It should be p o i n t e d out distribution any functions. that the G and F i n Lemma 4.3 need not be Nor do f and g have to be r e l a t e d to G and F i n p a r t i c u l a r way. Theorem WU7 (CRA): value Under WU, the f o l l o w i n g are e q u i v a l e n t f o r two p a i r s of and weight functions ( V A » W A a n < ) v * w ( B» g) with r e s p e c t i v e LOSUF C A and Cg: (a) (GCCERA) F o r any F e D _ , CCE.(F) < CCE_(F), J A B where and CCE (F) r e f e r to the CCE of F f o r ( v . , w . ) and ( v _ , w ) , 15 A A a D respectively. (GCPRA) r and T1 (b) CCE.(F) A D In a simple a s s e t z , where E ( z ) risky-asset tial wealth. portfolio > r, demands of let x A (v ,w.) A A set-up with safe asset risky and x be the r e s p e c t i v e c o n d i t i o n a l a and (v ,w ) D who have i d e n t i c a l i n i - D O 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 ) F o r any F e D j , C^(x;F) i s at l e a s t as concave i n x as Cg(x;F). In a d d i t i o n , each of the above c o n d i t i o n s i m p l i e s the f o l l o w i n g : (d) (GCERA) F o r any F e Dj, CE^(F) - 81 < CEg(F), - where CE^(F) and CEg(F) v refer (e) w to the CEs of F f o r ( ^> ^) (GPRA) In a simple a s s e t z , where E ( z ) demands of (v A portfolio > r, ,w.) A let and (v a n V w ^ ( B' g)» set-up with respectively. safe asset r x and x be the r e s p e c t i v e A B ii and risky risky-asset ,w ) who have i d e n t i c a l i n i t i a l ii wealth. Then x. < x „ . A B Proof: (b) in (a) -*• (b) •*• ( c ) : follows Suppose there e x i s t s H e Dj such that y than C^CyjH). C ( A h so t h a t , l from Theorem U 7 . 3 . ; H Then, ) there e x i s t h^ < Cg(y;H) i s more concave and q e ( 0 , 1 ) such t h a t Cl(h H) i ; f o r some 6 > 0, C (h +9;H)-C (h ;H) A 2 A 2 C (h ;H)-C (h -q9;H) A 1 A 1 > 1 ( 4 > 9 ) C (h ;H)-C (h -9;H) B 1 Recall that J C ( t ; H ) d [H-F] A 2 B 2 ^(h^m-c^h^H) > = ^— WU [(l-p)F+pH] | . A ( ' 4 ' 1 0 ) I n e q u a l i t y (4.9) yields I which C (h H) + \ C (h ;H) < ~ C ^ - q G j H ) A A i ; 2 + \ C (h +9;H), A 2 is JC <t;H)d[H-<^+^)] A ^{wu [(i- )(i6 A P h i + > /C (t;H)d[H-(^ „ A h i \\^-™ \^-*A\-^ k Since ,1, q e + .1 - 82 - \ \ ^ \ , \ + B ^ > 0. >|p-l at p = 1, it is implied that, f o r some p s u f f i c i e n t l y + wva-p)4\ Similarly, i inequality 5 ) h J[(i-p)4fi + K Let y J asset x B hl be o the _ ! = y r be t h e i r H). Also 1 l Note " that equivalent < ] ) + p H 2 ( 1 %t leads -p ) ( k - e k + 1 q 2 + 9 ) + to 1, H P ]- ^B^^^-^e+T^-e^l- l > 4 n < - > to wealth of WU. and WU,,. A B t (4.12) C o n s t r u c t a safe ^ _ and a r i s k y a s s e t z = -=-6, , + -=-6, , • 2 h /y 2 h ^ 2 L 1 respective Q L e t x. and A c o n d i t i o n a l demands f o r z ( c o n d i t i o n a l upon p and let e(l+q). M x H common i n i t i a l h +qh r-i 1+q Q p (4.10) WTL B + 2 close V" "t^ 1 x^ < y Q 1 a n < x,>. to ( 4 . 1 3 ) WU [(l-p)F . d x It 2 - V "V^ * 1+ 1 can be v e r i f i e d and ( 4 . 1 4 ) below, 2+pH] < W U [ ( l - p ) F A . e(l+q). M A •'o r that (4.11) and ( 4 . 1 2 ) are respectively: + x o ( ~_ r ) + H], (4.13) P 2 WU_[(l-p)F ~tpH] > W U _ [ ( l - p ) F . , ~ .+pH]. B y z B y r+x.(z-r) o o 1 L (4.14) 1 ; S i n c e 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 , A B x„ < y < x , contradicting condition B o A (c) lies •*• ( a ) : In l i g h t that WU. i s more GMRA than WU_. suffices Suppose G i s to prove that (c) imp- a simple wups of F to o WU. and s i n g l e - c r o s s e s F at x* from the l e f t . A concave to (b). of Theorem U 7 . 3 , i t A the above amounts As C . ( x : F ) i s at l e a s t as A i n x as C g ( x ; F ) , by Lemma 4 . 3 , C(x*;F) J[G-F]C (x;F)dx A > q Define F " = (l-a)F+otG. 0 = he ^A^ICPO ( x «| J[G-F]C (x;F)dx. p ) It follows = x (4.15) B F d that G F / y ; ) [ " ] " -/[G-F]C (x;F)dx A - 83 - C!(x*;F) < {-|[G-F]C (x;F)dx} A B C!(x*;F) n d A (a) a implies ^ - WU„(F ) a OL D •*• (d) and (b) -> (e) OC. d £ ''V* Cg(x*;F) With CI, CI > 0, t h i s . TTTT | 'IOFO' > 0. B Hence, WIL(G) > WU„(F). B B f o l l o w by d e f i n i t i o n . Q.E.D. Theorem LGU7 (CRA): tional (a) V The f o l l o w i n g are equivalent and Vg w i t h LOSUF C ( x ; F ) A and C g ( x ; F ) , A (GCCERA) F o r any F e D , T CCE . ( F ) J CCE_(F) r e f e r to for of LGU f u n c - respectively: < CCE_,(F), A a pair where CCE.(F) the CCEs of F f o r A and B , and A D respectively. o (b) C (x;F) ( C o n c a v i t y ) F o r any F e D j , is A at l e a s t as concave i n x as Cg(x;F). If, in addition, b o t h 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 , A above c o n d i t i o n s (c) (GCPRA) asset In z, simple where E ( z ) to: portfolio > r, let x set-up with and x be the r e s p e c t i v e A money risky-asset wealth. Then x safe asset r and B V . and V„ who have I d e n t i c a l A B 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 < x risky conditional demand of A the B are e q u i v a l e n t a then initial distri- a b u t i o n they are c o n d i t i o n a l upon. Each of (d) the above c o n d i t i o n s (GCERA) F o r any refer (e) to z, j ( (e) CE (F) a simple where E(z) portfolio > r, let and (d) < CEg(F), A the CEs of F f o r A and B, (GPRA) In asset F E D implies money with and x A tional) where CE (F) A and CEg(F) respectively. set-up x below: r i s k y - a s s e t - demand 84 - of safe asset r and risky be the r e s p e c t i v e (uncondi- V . and V_ who have A B identical B initial wealth. Then x. <x„. A Proof: Omitted s i n c e i t Theorem 4 i n Machina In this compensated gated if another. we f i r s t to lay comparative sion, it is the i n d i v i d u a l i n the sense of risk can to aversion be ground f o r helpful impose dominance. demonstrate GCCERA than B" <=> "A i s more GPRA than B " . dicts the but convinced When we assume utility serves As function), to stated concave measure belief on the they are linear C(x;F), is the of the same in of all, in First comparative about risk preference GCERA than aver- orderings, i n d i s t r i b u t i o n and interesting that consist- assumptions, B" <= "A i s we more Gateaux i n Theorem LGU7, i n x than C „ ( x ; F ) of fundamental differentials LOSUF utility degree equivalence it of contracompara- theory. than We are previously 'theory-independent'. called the the because underlying preference GCCERA and GMRA are more sense that Gateaux d e r i v a t i v e more finding is widely-held thus i n the "A i s This GCCERA and GMRA depends thought investi- Under these f a i r l y b a s i c that tive that continuity We then implies follows. study equiva- more GMRA than B" => "A i s more GCPRA than B" => "A i s casual see certainty demand. sense four p r o p e r t i e s ency with s t o c h a s t i c to as a meaningful transitivity, able i n one summarized namely completeness, were Also, the meaning of one i n d i v i d u a l being s p r e a d , as w e l l as r i s k y - a s s e t Our f i n d i n g order defined than another lent, the s i m i l a r to the proof of Theorem WU7. (1982a). section, more r i s k averse is B is (abbrevation function risk to aversion whose in obtain for f o r every F . - 85 - of concavity conditional more GCCERA than Vg i f f the lottery-specific degree its LGU, C (x;F) A sense. is more When the put functional differently, form of when C, i s C is identical distribution-free, at all a l l distributions, or conditional versions of r i s k a v e r s i o n w i l l reduce to t h e i r u n c o n d i t i o n a l c o u n t e r p a r t s . r a t i v e r i s k a v e r s i o n for t h i s Another sence of case i s c h a r a c t e r i z e d i n Theorem EU7. distinction between Theorem EU7 and Theorem LGU7 i s the conditional diversifer This the requirement because a l l EU maximizers are g e n e r i c a l l y We have known that WU i s and E U . few structures two that of inherently useful i n Theorem EU7. WU over a preference LGU so diversifers). on WU are in far This vain. theory i n t e r m e d i a t e (one does We s h a l l being not happen to have the that mean find when we wish to o b t a i n more s p e c i f i c s WU agents the if they share the same weight concave LOSUF w i l l be more r i s k The appeal of the when the problem i n v o l v e s decision (cf. between LGU risky-asset WU agents are additional functional form of WU For example, function. It suppose can be shown w i l l be more GCCERA. function, functional explicit or reveal the agent with more averse. specific Theorem WU6) is the of CRA. same v a l u e all that the agent whose weight f u n c t i o n decreases f a s t e r Similarly, ab- diversifers. Comparisions between Theorem LGU7 and Theorem WU7 however advantages also The compa- in form i s particularly o p t i m i z a t i o n as i n the p o r t f o l i o the study demand to be examined i n the next - 86 - of the evident choice n o r m a l i t y p r o p e r t y of section. 5 DECREASING RISK AVERSION AND THE NORMALITY OF RISKY-ASSET DEMAND WITH DETERMINISTIC WEALTH Section 3 is devoted to individual's risk aversion. investigated the turn one The back to questions studying In S e c t i o n 4, implications single of behavioral implications of an we compared two i n d i v i d u a l s and comparative individual we attempt the risk but a l l o w his aversion. initial to answer a r e : As an i n d i v i d u a l Now, wealth let us to v a r y . gets r i c h e r , will he be w i l l i n g to pay a h i g h e r or lower i n s u r a n c e premium f o r a g i v e n r i s k ? Will his demand f o r world i n c r e a s e or the that it the of the the decreasing wealth. maker has to be s t o c h a s t i c section risk this initial about we c o n t i n u e to wealth. We w i l l the assume allow section. two s u b s e c t i o n s . aversion assumptions In S u b s e c t i o n c h a r a c t e r i z a t i o n under E U . 5.1, we review Subsection 5.2 demand under WU. D e c r e a s i n g R i s k A v e r s i o n under Expected U t i l i t y Arrow lute section, deterministic i n the next contains In focuses on the n o r m a l i t y of r i s k y - a s s e t 5.1 one-safe-asset-one-risky-asset can be answered under d i f f e r e n t agent's decision This i n our decrease? These q u e s t i o n s riskiness r i s k y asset risk become (1971) c o n v i n c i n g l y argued f o r a v e r s i o n which i m p l i e s less risk averse in that, the as sense - 87 the decreasing an agent gets r i c h e r , of - appeal of demanding cheaper abso- he should insurance policies If as if risk and i n v e s t i n g the In same i n d i v i d u a l with a d i f f e r e n t he were a d i f f e r e n t aversion creasing can more money i n the r i s k y a s s e t . be done. accordance (DRA). straightforwardly EU i s a preference Theorem EU8 below i s a direct with we by the the of wealth can be then the c h a r a c t e r i z a t i o n of (CRA) can be r e s t a t e d r i s k aversion conditions person, level literature, to comparative characterize de- theory under which translation replace viewed the of this Theorem EU7. certainty equivalent i n s u r a n c e premium ones and s t a t e the c o n c a v i t y condition i n terms of A r r o w - P r a t t i n d e x . Theorem EU8 (DRA, A r r o w - P r a t t ) : The f o l l o w i n g increasing, t i o n u(y) twice-differentiable are von properties of a continuous, Neumann-Morgenstern u t i l i t y func- equivalent: u"(y ) u'Cy^ > - —r-. r- f o r a l l y < y. • u'(y ) u'(y ) o '1 Q (a) ( A r r o w - P r a t t Index) - r Q (b) (Conditional pose n o = i t ( y , e P,H) premia respectively. (Conditional safe are and (d) asset u's y^, for Then u Premium) For any p e (0,1] and H e D , = n:(y^, e j p , H ) and o insurance (c) Insurance 1 risk > Risky-Asset r and r i s k y e at for a l l Demand) asset (Insurance insurance Premium) premia respectively. for Then X q Suppose risk Then % > e if In z, conditional risky-asset respectively. initial wealth Q u's conditional levels y a simple where E ( z ) < x^ i f y Tt portfolio > r, set-up suppose X q levels initial - y Q < y^. Q o e if with and x^ = n ( y , e ) and %^ = n:(y^,e) are for a l l 88 and y^, Q < y^. demands at i n i t i a l wealth e at - y are sup- y wealth Q < y^• levels y Q u's and y^, (e) (Risky-Asset and risky asset Demand) In a simple a s s e t z , where E ( z ) demands Then x o at < x. i f 1 y Theorem EU8 says decreasing ditional, al, absolute o initial > r, levels richer. l i t e r a t u r e as the that, if asset then h i s wealth and the preference appears forward wealth we continue increment functionals, that, implication is how are will often to assume of its CRA c o u n t e r p a r t s . the that but characterization beyond E U , the DRA c h a r a c t e r i z a t i o n s restatement referred to as he in the following two demand'. deterministic the condition- demand w i l l i n c r e a s e a benchmark, we may c o n s i d e r if exhibits c o n d i t i o n a l , as w e l l as uncon- any r i s k w i l l decrease and h i s demand Taking Theorem EU8 as First, risky- respectively. an EU d e c i s i o n maker's p r e f e r e n c e ' n o r m a l i t y of r i s k y - a s s e t generalizations. and y^ , Q r < y, . •'I premium f o r The y asset and x^ are u's q as w e l 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 gets s e t - u p with safe suppose X wealth risk aversion, insurance portfolio both the initial adopt more general be modified? w i l l not be a This will be It straightexplained shortly. The wealth or second even generalization the T h i s w i l l be d e a l t 5.2 wealth of Theorem EU8 i s increment with i n S e c t i o n from to extend deterministic to the initial stochastic. 6. D e c r e a s i n g R i s k A v e r s i o n and the N o r m a l i t y o f R i s k y - A s s e t Demand under Non-Expected U t i l i t y Once arise in the utility directly function translating becomes lottery-specific, CRA c h a r a c t e r i z a t i o n s - 89 - to difficulties DRA ones. To illustrate, under WU. initial Cg, consider In the the unconditional context wealth y . J of CRA, suppose L e t x. and x „ denote A B o respectively. simple portfolio both choice investors the r i s k y - a s s e t have problem the demands of J same C» and A By o p t i m a l i t y , we have JC.(y r+x.(z-r);F A o A y . , ~ .)dF(z) = 0 r+x,(z-r) A 1 J o J o (5.1) and In the context B 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 ^ ,~ )dF(z) = 0 . A l l ' y^r+x^(z-r) ' J w v / v v C at y.. can be viewed as i n v e s t o r A i If investor by Cg and y^ by y J N i n (5.3) Q to o b t a i n ( 5 . 4 ) I C , (Jy r + x . ( z - r ) ; F . , ~ JdF(z) B o 1 y r+x, ( z - r ) o 1 C at y , we may r e p l a c e C B o A below: = 0. v (5.3) ' v (5.4) 7 y The only CRA c h a r a c t e r i z a t i o n i n Theorem WU7 can be if the tween x, r e l a t i o n between Xg and y and y 1 that in (5.4). This upon • in is c h a r a c t e r i z e DRA i d e n t i c a l to that would be the case i f (5.2) ' . , ~ v ) were i d e n t i c a l . y r+x,(z-r) J o 1 (i.e. be- the d i s t r i b u t i o n s DRA c h a r a c t e r i z a t i o n i n terms of ply by r e p h r a s i n g the applies does characterizing asset F ,~ *) y r+x_(z-r)' o B S i n c e x „ + x, B 1 the Why i n (5.2) to •'o C„ d e p e n d s B ment a l s o Q used asset and ( 5 . 4 ) ' v v (i.e. i n g e n e r a l , we conclude that demand cannot be o b t a i n e d s i m - demand c o n d i t i o n i n Theorem WU7. This argu- to the i n s u r a n c e premium c o n d i t i o n . the DRA? distribution-dependence In the context - of 90 of LOSUFs CRA, say - in cause terms problems of in insurance premium, C^(y;6 compare the y o because degree ) and Cg(y; 6 _ y our we the __) B o distributions interest to of concavity of two individuals' LOSUFs which are exogenously g i v e n and do not they DRA, a g a i n , depend upon remain unchanged. we have two LOSUFs, shift As we one depends turn on F ~, y another other on F y because change of depends an However, they originate individual's on the explained ~ . + movement i n greater these two LOSUFs are not from risk along detail the same attitudes £ as after well c o n d i t i o n f o r the n o r m a l i t y of r i s k y - a s s e t One enable advantage us without results Theorem to WU over optimize utilizing directional WU8.1 (DRA and of a pair with i n c r e a s i n g , E (a) p(y;p,F,H) is (b) using LGU i s its calculus or path his varies wealth shift the of . necessary The therefore This and each will be sufficient demand under WU. structural and specifications perform comparative differentiation. which statics This advantage The following and weight functions i n Theorems WU8.1 and WU8.2 below: properties (v,w) of functional. the we d e r i v e of preference as as independent o = decreasing Conditional of x^ be concave LOSUF C ( x ; F ) are C"(y;G) _ pw'(y) [C'(G)] E [w] C'(y;G) (where G = p F + ( l - p ) H ) i n y f o r a l l p e (0,1] Demand) equivalent: conditional and y , 1 In a simple risky-asset respectively. - Then, x 91 (5.5) and F , H e D^; portfolio set-up a s s e t r and r i s k y a s s e t z = r+n, where E ( n ) > 0, (v,w)'s levels y Demand): properly structured value (Conditional Risky-Asset riskfree Risky-Asset - demands < x. if at y let initial < y,• with X q and wealth Proof: The CSPC under WU i s Max V(pF x ^o r + x The ^-(l-p)H). FOC and SOC f o r x* to be o p t i m a l are as FOC: /C'(y*;G)TtfF(Ti) SOC: /C"(y*;G)T) dF(Ti) follows: = 0 (3.9) < 0, 2 (3.10) where G = p F ~ + ( l - p ) H and y* = y r+x*r). !fc The rest of o the proof i s s i m i l a r to that of Theorem WU8.2. Q.E.D. Although to present Theorem WU8.1 i s a complete notation-wise proof more g e n e r a l of and s a c r i f i c e s the little latter than Theorem WU8.2, because it is less of properly increasing, structured (a) [C - - P^y,*; - (b) F £"(y» > c'(y;F) decreasing risky 1 functions (v,w) a with 7 i V g ] (5-6) + + w'(y)/E[w] ( C'(y;F)/E[C(F)] ? . in y for a l l F; asset risky-asset ly. weight of equivalent: ( R i s k y - A s s e t Demand) In a simple and Proof: and properties equivalently, ofv-m is complicated _ w'(y) p(y;F) * ~ * * ™ \ or value concave LOSUF C(x;F) are C"(y;F) elect substance. Theorem WU8.2 (DRA and R i s k y - A s s e t Demand): The f o l l o w i n g pair we z = r+n, demands at Then, x < x. i f ' o 1 initial y J where o portfolio E(r)) > 0, wealth < y, . 1 J The SPC problem under WU i s - 92 - s e t - u p with safe asset let (v,w)'s levels y X Q q and x^ be and y^, r respective- Max J jv(y)w(y)dF ~ ~ r ' — — , where y = y r+xn. Jw(y)dF ' o J J The FOC and SOC f o r x* to s o l v e FOC: /C'(y*;F)TidF(Ti) SOC: /C"(y*;F)T dF(T ) follows: = 0 2 Implicitly differentiating (5.8) < 0. 1 1 the above are as (5.9) the FOC ( 5 . 8 ) w.r.t. x* and y Q yields ,j r F)-.'(irgjg) *F [ ( 1 _ y where SOC, 2 the argument the (5.iu; t -/C(F)Ti dF o y has denominator dx* s i g n of —z o is the of been the omitted RHS of same as that to simplify (5.10) is the positive. expression. Therefore, By the of d y , J[C '(F)-w'( / e , d F a> )]TidF TwdF / which can be r e w r i t t e n EtC"(y;F)-^] as - E [ F ^ ) E[W'(?')'?I]. ] (5.ii) W M J Also, (a) r e c a l l that E ( z ) •*• ( b ) : > r i m p l i e s x* We need to c o n s i d e r Case ( i ) : > 0 a c c o r d i n g to Theorem U6. two cases — n > 0 and n < 0. n > 0 y = y r+x*Ti > y r . Q Q E x p r e s s i o n (5.7) _ f decreasing C"(y;F) _ E [ C ( F ) ] C'(y;F) E[w] l Multiply implies that w'(y) > C'(y;F) = J p ( y r o ' F ) ~ p o' both s i d e s by - C ' ( y ; F ) T i : C"(y;F)n Case ( i i ) : - w'(y)n E [ ^^ ) ] > -C'(y;F)np . o n < 0 y = y r+x*n < y r . o Expression (5.7) Q decreasing implies that (5.12) l C'(y;F) Multiplying E[w] C'(y;F) 1 p o' both s i d e s by - C ' ( y ; F ) n y i e l d s so t h a t we can take i t s e x p e c t a t i o n as E[C"(y;F)^i] - E[w'(y)r^] E [ ) ^ f ? a [ wj inequality follows: > ] (5.12) w i t h s t i c t -p E[C'(y;F)9i] = 0. o (by FOC) dx* Hence, -=— > 0. (b) •* ( a ) : tial To prove n e c e s s i t y , wealth level y . Then, Q suppose (5.7) is i n c r e a s i n g at some f o l l o w i n g steps i n the s u f f i c i e n c y iniproof dx* will l e a d to —— < 0, d y which c o n t r a d i c t s c o n d i t i o n (b). o Q.E.D. The l o g i c (Theorem i n the proof EU8). portfolio show t h a t We f i r s t choice totally p r o b l e m to p g i v e n by ( 5 . 6 ) Theorem WU8.2 i s following of Theorem WU8.2 i s differentiate obtain the discussion w i l l focus the FOC f o r zation to Arrow (1971). from Arrow's r e s u l t In c o n t r a s t , e q u i v a l e n t to d x * / d y case of Theorem WU8.1. functional, Within By ' o n e - s t e p ' the other i s the simple Then we Q > 0. Nevertheless, on Theorem WU8.2 because it is a one-step demand' under EU the p r e f e r e n c e two changes — one is functional. the preference 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 . domain of EU, Arrow (1971) showed that, when the initial w e a l t h 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 ference will single risky d i s p l a y decreasing absolute asset is the we mean that the o n l y g e n e r a l i - to Theorem WU8.2 i s Theorem WU8.1 i n v o l v e s the Q e x t e n s i o n of the now well-known ' n o r m a l i t y of r i s k y - a s s e t attributed i n Arrow's e x p r e s s i o n of d x * / d y . decreases i n y i s a special s i m i l a r to that risk aversion i f a normal good to him ( i . e . - 94 - dx*/dy and o n l y i f Q > 0). prethe In a d d i - tion, his preference w i l l only if 0). The former r e s u l t the d i s p l a y i n c r e a s i n g r e l a t i v e r i s k a v e r s i o n i f and safe a s s e t i s is a s u p e r i o r (or l u x u r y ) good ( i . e . c o n t a i n e d i n the equivalence of d(l-B*)/dy (c) > Q and (e) in Theorem EU8. It may at first appear somewhat surprising that condition v"(v) nor the c o n c a v i t y index - C " ( y : F ) / C ( y ; F ) . of WU, p(y;F) must reduce to is constant. constant, Pratt It can be v e r i f i e d neither index will a decreasing imply or be p a r t l y e x p l a i n e d at the s t a r t that this concavity p(y;F) does n o t . vestor the This depends risky for i t , depends of t h i s asset. recall To see risk a special indeed nor so. the final a decreasing p(y;F). of simple portfolio wealth y = y r+x(z-r). Q r e g r e t when v ( y ) d e r i v a t i v e of We have asset the is attitudes Arrow-Pratt index note r(y) a normal good to a WU i n but a l s o the a t t r i b u t e s of one's demand that context simply c a l l not Arrow- As to the A r r o w - P r a t t i n d e x , how the d i s t r i b u t i o n of z a f f e c t s WU(F) w i t h w(y)//wdF being us case When w i s C(y;F) = w ( y ) [ v ( y ) - W U ( F ) ] / / w d F . In J s u b s e c t i o n why a d e c r e a s i n g c o n c a v i - means whether a r i s k y only h i s w(y) index - v " ( y ) / v ' ( y ) when w index on d i s t r i b u t i o n F while on not v'(y) i m p l i e d by a d e c r e a s i n g t y index i s not the r e q u i r e d c o n d i t i o n . that is 2w'(v) s Given that EU i s the EU A r r o w - P r a t t in ,y' + — 7 ^ / - ] Theorem WU8.2 i s n e i t h e r the WU A r r o w - P r a t t index r ( y ) = - [ J (a) the (1.25) choice, C(y;F) i s weight. < WU(F) and a weighted F is We may i n t e r p r e t C as r e j o i c i n g when v ( y ) C g i v e n below: 95 d i s t r i b u t i o n of the a weighted u t i l i t y - d e v i a t i o n from C (or more s u i t a b l y - £ ) a weighted - the - regret a weighted > WU(F). Let in general. The w' ( y ) [ v ( y ) - W U ( F ) ] + w ( y ) v ' ( y ) /wdF C'(y;F) = - is the contingent contingent by $1, by 7 5 ^ it [v(y)-WO(F>] marginal marginal causes two e f f e c t s [w'(y)/E(w)][v(y)-WU(F)] one is weighted weighted a 'utility [w(y)/E(w)]v'(y) u t i l i t y - d e v i a t i o n from WU(F) or regret for on £ . The f i r s t — marginal effect', (1.32) + ^ V ( y ) given — a weighted m a r g i n a l the outcome y . is weight by When y a 'weight times increases effect', regret. the given The second [w(y)/E(w)][v(y)-WU(F)]' = regret. The FOC f o r a WU m a x i m i z e r ' s SPC problem i s (5.8) /C'(y*;F)TidF(n) = 0, or e q u i v a l e n t l y , (5.8') For an a d d i t i o n a l d o l l a r ' s investment z realizes. t i o n of 'bad' from z. C'(y*;F)r) i s The LHS of outcome 'good' states outcome i n z, the e x t r a income i s the m a r g i n a l u t i l i t y c o n t i n g e n t (5.8') while gives the states. T) = z - r i f on the r e a l i z a - the expected m a r g i n a l d i s u t i l i t y from RHS g i v e s The FOC means the expected that, at marginal optimality, utility the ex- pected m a r g i n a l u t i l i t y and d i s u t i l i t y from i n v e s t i n g an a d d i t i o n a l $1 i n z must b a l a n c e out so that the agent has no i n c e n t i v e to d e v i a t e risky-asset Note y . 'o demand x*. that in Therefore, increases, from h i s (5.8) i n o r d e r to C'(y;F) C'(y;F) w . r . t . y C'(y;F) Q must yields the only term i n v o l v i n g the parameter have the r i s k y - a s s e t behave the is in a particular following: - 96 - demand x* r i s e when y manner. o Differentiating W~ C ( y , F ) " C C y , F ) "JwdF" = r{c"(y;F) - iW- /C'(F)dF} = r{C(y;F) E [ C ' ( F ) ] }. TwdF The interpretation of - (5.13) p(y;F) defined by (5.6) is better illustrated by expressing i t as follows: [-§£- C ' ( y ; F ) ] / r E [ C ' ( F ) ] P(y;F) = - ° C'(y;F) rE[C*(F)] {_ £Snll I E[C [_£- C ( y ; F ) ] / V ( y ; F ) 1 '(F)] + C'(y;F) iM^fl^I} + C'(y;F) ( extra = dWU(F)/dy dollar dominance, available it must is o for be the M 6 ; C"(y;F) _ w'(y) _ _ E[C'(F)] E[w] C'(y;F) rE[C'(F)] 5 >* ( 5 ' 6 ) ex ante expected m a r g i n a l u t i l i t y from an investment. positive. To be c o n s i s t e n t Since rE[C'(F)] is with constant stochastic w.r.t. y, p(y;F) w i l l be d e c r e a s i n g i n y i f and o n l y i f - C'(y;F)]/C'(y;F) y is also (5.14) o decreasing i n y. y -elasticity, i.e. increase i n y . Note that an i n c r e a s e i n y WU(F), the Q Q changing measured. Thus, movement along the E x p r e s s i o n (5.14) proportional benchmark based the e f f e c t C'» the other has change Q the in will from the twofold, shift C'(yjF) a i n d u c e d by $1 cause an upward s h i f t on which the on C ' ( y ; F ) i s i n t e r p r e t a t i o n of magnitude of regret of is one r e s u l t i n g from the of WU(F). These two effects are r e p r e s e n t e d i n (5.13) by the two a d d i t i v e terms i n the c u r l y b r a c k e t . - 97 - p decreasing marginal y. of i n y means weighted r e g r e t that the 'normalized' y - e l a s t i c i t y Q of the ( n o r m a l i z e d by oWU(F)/oy ) must be d e c r e a s i n g i n o A decreasing y - e l a s t i c i t y Q i n t u r n means t h a t the i n v e s t o r ' s intensity r e g r e t about not o b t a i n i n g a m a r g i n a l l y b e t t e r outcome s t a t e l e s s e n s he becomes r i c h e r . T h i s seems reasonable to be the c o n d i t i o n f o r the m a l i t y of r i s k y - a s s e t of the r i s k y How must to demand — as an agent gets r i c h e r , decision nor- he w i l l h o l d more asset. the value display decreasing question, as f u n c t i o n v and weight y -elasticity in Q f u n c t i o n w behave C'(y;F)? i n order Before we answer assume that the SPC problem i s u n i q u e l y s o l v e d f o r a r i s k maker, C(y;F) i.e., > 0 and C"(y;F) < 0. The f o l l o w i n g this averse table should be u s e f u l . (a) (b) (c) case -C/C w w" (1) 4- 4- - (2) 4- 4- + (d) (e) w'/C condition for (w'/C)' < 0 c o n d i t i o n f o r p' < 0 - always 4- ? w - " < - £.* w' ^ (3) 4- + (4) 4- t (5) t 4- + (f)*** rill' C ? - ^-r > - 4r-* w' C + impossible + - l t 4- + ? (7) t + - ? (8) + + + + impossible * s u f f i c i e n t but not necessary f o r p' < 0 ** n e c e s s a r y but not s u f f i c i e n t f o r p' < 0 98 > J W/E[w] ,, C(F)/E[C(F)] ' l - above as above -rill- < -\ w" C"** - —r < - -5-r w C, w" £"** - —r > —=r w C - J as L (6) C C J W '/ r' E[w] C(F)/E[C(F)] as above l as above impossible * * * independent of (e) J In the above t a b l e , i n c r e a s i n g l y concave. of decreasing r i s k tural case. increasing concave as (5.6)) w i l l direction sing i s cases is as and (3) In case by the agent's but result. It interesting GMRA as he gets r i c h e r . asset weight reduces category p (cf. function is is and expression For a l l other c a s e s , is i n y, the be d e c r e a - case (1) p will display reduce h i s or still that even i f but in in z When - w'/C' de- the n o r m a l i t y the LOSUF C, i s increasing- attainable. C means that the agent w i l l How can we j u s t i f y us (e). aversion investment increases -C"/C (2) i n column increasing risk he w i l l definitely as w e l l as cases a d d i t i o n a l condition given to note when he never occor under E U , l e t the function decreasing term must dominate i n order to have normality is risky and ( 8 ) . demand, i . e . An i n c r e a s i n g l y concave the term of decreasing w'/C' i n c r e a s e s , the d e c r e a s i n g of the weight When w i s a form the more n a - w e a l t h i e r — a case n o r m a t i v e l y not very a p p e a l i n g . creases, concave, whether second (4) preferences risky-asset decreases is demand i s concave LOSUF i s convex. the d e c r e a s i n g l y or (e). when r e s t r i c t e d of £ is The c o n d i t i o n under which w ' / £ ' w i l l In t h i s sense whether On the c o n t r a r y , i t w i l l be i n c r e a s i n g i f w i s i n y as w e l l . as he becomes ly (5), and w'/C' are -C'7 C' the or i n cases ambiguous. (8), indicate concave and g i v e n i n column decrease C"/C (1) and convex a decreasingly and (c) be d e c r e a s i n g . When both the aversion, decreasing, in specifies Since the n o r m a l i t y of r i s k y - a s s e t Columns (b) or increasing column (a) has more money such an a g e n t ' s demanding more to invest? c o n s i d e r an EU agent constant, the second to - 99 - be more GCCERA or Since this as our base c a s e . term of p vanishes will When and p v"(y)/E[v'] v'(y) " Since E[v'] decreasing is is constant i n y. i n y, The l a t t e r a normalized concavity w i t h the A r r o w - P r a t t When w i s p decreasing EU not is index in y is equivalent of course the EU A r r o w - P r a t t whose behavior happens to to - v " / v ' index. be consistent index. constant, it is clear from (5.6) that a concave and/or d e c r e a s i n g w w i l l r e i n f o r c e the d e c r e a s i n g r i s k a v e r s i o n captured by a decreasing normalized concavity index. creasing w w i l l offset dicates sing agent a l l or p a r t of pessimism which i s w therefore whose On the c o n t r a r y , a convex a n d / o r i n - depicts a source it. of risk decreasingly utility-based risk R e c a l l that a d e c r e a s i n g w i n aversion. pessimistic aversion is attitudes. increasing [-C'VC]' > 0 ) » If h i s pessimism decreases s u f f i c i e n t l y comes r i c h e r , he might s t i l l Interestingly, if we increase define his a A concave, in decrea- For a WU wealth fast (i.e. when he be- h o l d i n g i n the r i s k y a s s e t . proportional version of p(y;F) as follows: yC"(y;F) _ yw'(y) P*(y;F) it can be good) i f Can = yp(y;F) shown we o b t a i n is E[w] (5.15) C'(y;F) d(l-B*)/dy > 0 (i.e. Q p*(y;F) i n c r e a s e s a similar of that for the weight this the safe a s s e t i s is the normality that the of not p o s s i b l e without 100 - apart from imposing more assumptions. i n s u r a n c e premium c o n d i t i o n i s - risky-asset function i n (5.6) on the f u n c t i o n a l V(F) or making f u r t h e r worth n o t i n g a superior i n y. condition The presence l e a d s us to b e l i e v e structures It that and o n l y i f demand under LGU? C(y;F) E[C(F)] absent in Theorems WU8.1 and WU8.2. tions that between the he experiences as altered. not an be some fundamental d i s t i n c - i n the risky the cannot wealth in his asset. changes. s i z e of be sure that all, note the In c o n t r a s t , investible investible Suppose h i s risky-asset absolute of i n s u r a n c e premium c o n d i t i o n r e - a f u n c t i o n of h i s increase First initial size he As will wealth i n - from X to q the r i s k he i s relative wealth. funds, demand changes an x^. bearing w i l l of the r i s k be will vary. a as p o i n t e d out e a r l i e r , decision risk. In behavior. use to i n the initial Q We a l s o Secondly, of agent's exogenous * x^ , q risk to y +Ay and h i s Q X the demand i s investment c r e a s e s from y long of risky-asset change h i s As size same when the investor's appears i n s u r a n c e premium and p o r t f o l i o c h o i c e . the a b s o l u t e mains There of maker's contrast, As such, the FOC i f it perception about risky-asset the demand d e r i v a t i o n of is i n s u r a n c e premium i s obtainable. the certainty is the the a consequence equivalent result of c o n d i t i o n for for a an o p t i m i z i n g PRA o f t e n makes We have a p p l i e d the same approach i n p r o d u c i n g Theorems WU8.1 and WU8.2. A natural insurance way of premium i s t i a t i n g WU(F + ~) deriving the condition comparative s t a t i c = WU(6 o the = v(y -ix) o for DRA i n technique. the Totally sense of differen- yields o JC'(F. v ' ( yJ DRA i n the sense of f /C (F J J (5.16) -it) i n s u r a n c e premium c a l l s ~)dF y +e o o '~ y +e o J > v'(y w o < 0, i.e., (5.17) -n). - f o r dii/dy 101 - Under EU, (5.17) reduces to Ju'dF J ~ y +e o which ference holds > u'(y -it) w o f o r a l l e i f and only i f -u"/u' i s d e c r e a s i n g . functional i s nonlinear i n distribution, v a l e n t to (-C/C')' < 0 i n g e n e r a l . - 102 - (5.17) w i l l When the p r e not be e q u i - 6 COMPARATIVE AND DECREASING RISK AVERSION INVOLVING STOCHASTIC WEALTH In assumed Section 5, It is in Theorem asset therefore It does not of portfolio > 0 be wealth w i l l the two assets final When Ay i s is caused a investible, 'resource resources. we assume the a v a i l a b l e or extended to allow for change risk wealth increment may confound To i l l u s t r a t e , let initial the r and z = r+n w i t h simple p o r t f o l i o set-up. wealth i s Suppose an y . His final to x, = x +Ax (Ax may 1 o J wealth i n c r e a s e s o o to y^ = y +Ay Q Ay i s investible, investible. a consequence — an e f f e c t i n his i n his attitude anticipated stochastic increments. of caused two e f f e c t s . attitudes 'consumable' effect wealth - 103 in this in his effect' — toward r i s k which i n t u r n income. We s h a l l section. increment - One may by an i n c r e a s e The other may be c a l l e d a ' r i s k a t t i t u d e by an i n c r e a s e that not are deemed u n r e a l i s t i c . be (y +Ay)r+(x +Ax) T) i f Ax i s effect' caused by the concerned with the the i n our Ay i s not increment how the CRA and DRA c h a r a c t e r i z a t i o n s that wealth w i l l Q an e f f e c t assumption i s wealth when h i s wealth insurance i s h i s demand f o r z changes or n e g a t i v e ) his and the when h i s r i s k y i n i t i a l If and y r+(x^+Ax)n+Ay i f investible see c h o i c e problem. be Jy r+x n. o o positive called this or even s t o c h a s t i c q be to should be p o i n t e d out (Ay > 0 ) , wealth When complete exist, interest a g e n t ' s demand f o r z i s X be initial EU7 and Theorem EU8 can be wealth, agent's E(n) the to be d e t e r m i n i s t i c . when a safe initial both will only be In other words, be available only after the plicity, tricted Note investment decision we say the wealth wealth that, increment when wealth the safe a s s e t , This is made, increment i s in increment contains is is ex p o s t . ex ante section two p a r t s . Theorem EU10, due to Ross (1981), pectively, 6.2, we and WU8.2 For s i m the u n r e s - is ex ante. but can o n l y be i n v e s t e d holds. In S u b s e c t i o n 6 . 1 , initial wealth. Theorem EU9 and first In Theorem EU10, which assumed d e t e r m i n i s t i c . illustrate that index does not have a WU or LGU c o u n t e r p a r t . Ross' strong concavity Theorems LGU9 and LGU10 are then presented as s p e c i a l cases of Machina (1982b)'s Theorem 1 extended LGU by imposing (Chew, tic 1985). wealth allowed additional Again, structure we assume increments. to be s t o c h a s t i c initial (1981) or based discussed the the results completeness gathered on Machina but not this This wealth among are this E U , WU and namely from ( y , A y ) , Q (y ,Ay). Q - 104 - are no Theorem EU11. and Chew (1985), of increment case a p p a r e n t l y i n v o l v e s section The presence comparisons assumptions about wealth l e v e l s , to in wealth and d e t e r m i n i s - and Hence, there i s (1982b) reproduced. of wealth to Gateaux d e r i v a t i v e C linear initial i n Theorem LGU11. too many r i s k s f o r EU to h a n d l e . Since on the stochastic Both in extend Theorem EU7 and Theorem EU8, r e s - to a l l o w for s t o c h a s t i c Subsection In c o n t r a s t , still i n terms of DRA, the wealth increment i s In noninvestible. Theorems EU8, WU8.1, the r e s u l t i n t h i s section therefore either from Ross t h e i r proofs w i l l section i s mainly f o r LGU under to ( y , A y ) , Q be different and then 6.1 CRA and DRA w i t h S t o c h a s t i c Wealth under Expected U t i l i t y Suppose y . Let H two EU a g e n t s and a be t h e i r r e s p e c t i v e c o r r e c t e d with y . in a simple E(rf) be willing Let x portfolio > 0 for all to given risk asset than u y A If w i t h a s s e t s r and z = u^ i s more r i s k averse a higher risky-asset r+ri, e un- demands where E ( r f | y ) = Q than Ug, we expect u^ to premium t h a n u f o r the i n s u r a n c e a g a i n s t B S i m i l a r l y , we a n t i c i p a t e will. wealth i n s u r a n c e premia f o r a r i s k and Xg be t h e i r r e s p e c t i v e set-up . pay e. u^ and Ug have the same r i s k y i n i t i a l What i s the Ug to proper invest more i n the c o n d i t i o n f o r being a risky 'more r i s k A. averse' in this sense? The answer i s Theorem EU9 (CRA w i t h continuous, y strictly Morgenstern u t i l i t y , Ross): g i v e n i n Theorem EU9 below: The f o l l o w i n g increasing, functions u properties twice-differentiable of a p a i r of von Neumann- and Ug are e q u i v a l e n t : u (y+k) Ug(y+k) - —, , > - —, , . u (y) B A A (a) (Strong A r r o w - P r a t t Index) (6.1) N u ( y ) A for a l l (a') k. (Strong C o n c a v i t y ) There e x i s t a p o s i t i v e s i n g concave u (y) A (b) f u n c t i o n h such \ f o r I n a d d i t i o n , each of (c) that (6.2) Suppose it and % are the r e s p e c t i v e insurance A B 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 . A B o r > decrea- Premium) premia f o r r i s k \ \ and a = ^Ug(y) + h ( y ) . (Insurance Then, constant a 1 1 ^ satisfying E(e|y ) Q the above i m p l i e s ( R i s k y - A s s e t Demand) Suppose u - A the f o l l o w i n g and u 105 = E(e) for a l l - f i y . Q condition: have i d e n t i c a l wealth y Q and x , X g are t h e i r r e s p e c t i v e lio s e t - u p with a s s e t s r and z = r + T ) , where E ( r j j y ) = E(t)) all y . o A insurance A > 0 for Then, x. < x „ . ' A B 3 the r i s k y - a s s e t to the other c o n d i t i o n s . case, x demands i n a simple p o r t f o Q Note that lent risky-asset demand c o n d i t i o n i s i m p l i e d but not e q u i v a - T h i s i s another evidence premium and r i s k y - a s s e t demand i s not that quite the the nature of same. In this < X g implies 0 > EIuj^r+XgTOri] = E[Xug(y r+XgTi)Ti] o + E [ h ' ( y ^ r + j ^ T ) ) TI] = E [ h ' ( y r+xJn)TJ] D O - C o v [ h » ( y r+x ^),^] + E[h» (y r+xJn)]E[^n] O a D O which may be s a t i s f i e d by a f u n c t i o n h not simultaneously d e c r e a s i n g and concave. The s t r o n g A r r o w - P r a t t index c o n d i t i o n i n Theorem EU9 i m p l i e s , not i m p l i e d by, its stronger randomness the Arrow-Pratt form h e r e , is the To g a i n some i n s i g h t wealth y . ly not Moreover, The (6.3) risk let insurance if y e is Q c o n d i t i o n i n Theorem EU8. wealth It i n t o Ross' is allowed risk asset. on the r e a l i z a t i o n of Q o 2 contingent below: - 106 - Q risk the and y + £ > Q is initial definite- f o r every y o F and G be the d i s t r i b u t i o n s of y c o n s i d e r an e Y and w i l l e Y; E ( e | y ) = 0 and V a r ( e ' j y ) = a this to must be u n c o r r e l a t e d to the l i k e l y to occur o n l y i f y premium it f o r Due The strong Arrow-Pratt index, contingent is to be random. n from the r i s k y e which i s Suppose occur initial however not a r b i t r a r y . e to be i n s u r e d or the r i s k infinitesimal index but defined Q e Y. respectively. by equation Ju(y -n)dF = J u(y +E)dG o + O Y /_u(y )dF. Expand both s i d e s v i a T a y l o r ' s s e r i e s as Ju(y -Tt)dF = follows: 2 J[u(y )-mi'(y )+0(Tt )]dF o o o 2 = J u ( y ) d F - n;Ju'(y )dF + 0 ( n ) . Q / u(y +£)dG + Y (6.3) Q o J_u(y )dF o o = J [u(y )+eu'(y )+ ^ = J u(y )dF + ^ = Ju(y )dF + ^ Y o y 2 u"(y )+o(e )]dG o Q J u"(y )dF + o Y o / u"(y )dF + Q Y Q + J_u(y )dF Q 2 / _ u ( y ) d F + o( a ) Q 2 o(a ). Therefore, a V'< o -T-Ju'(y )dF 2 y e ) * o» The utility risk the term £ will / u"(y )dF Y r no The matter $1 £ Y, Q uninsured loss. Since Q chastic, and has o J index the conditional £ Y or y o e Y. expected i n premium f o r the i n s u r a n c e . f o r the case interpretation v a r i a n c e " of The the of r a t i o of these two where "twice the the i n t e g r a l i s taken over term - / u ' ( y ) d F on the other hand g i v e s the y J 4 ) d i m i n i s h i n g r a t e of the m a r g i n a l from extra d o l l a r - The i s taken over the union of Y and Y because the premium has Arrow-Pratt of disutility be p r e s e n t o n l y at y modified unit ( 6 ' i s an expected the from paying one expectation paid o likely set Y only. ) d F o which measures disutility be " y the agent's the infinitesimal to terms i s the wealth insurance contingent i s sto- premium risk to per be insured. For two EU maximizers u. and u_, w i t h i d e n t i c a l y A B o have %^ > %^ f o r a l l such r i s k s o n l y i f J we - 107 - distributed as F, * d W V F « > /u (y )dF A for all Y. V { b /u (y )dF Q B Q F u r t h e r m o r e , under EU the r i s k e can be g e n e r a l i z e d finitesimal ones. It F ~ and Y i f o and o n l y is s t r a i g h t f o r w a r d to check that (6.5) 5 - > to n o n i n - holds for all if y u (y+k) u (y+k) A B >_ ~ u (y) -sp?r (6,1) A for all k. Condition (a') u via (6.2). of Theorem EU9 says u Recall that, if U obtained by "concavifying" can be obtained A u i s more concave A u^ v i a an i n c r e a s i n g , (6.2) r e q u i r e u be even more concave? A t i o n u „ , suppose by t r a n s f o r m i n g than u , then u can be 15 A concave function. G i v e n an i n c r e a s i n g , concave Does func- D u (y) = Au (y) + u (y) » h[u (y)], A h(y), B and A X > 0 , h' where u , A B u A < - X u l < 0 , h" < 0 , h* > 0 and h" < 0 . B - < 0 and U A characterization initial wealth, / U > A - U U A^ A* n * o t ^ e r words, i n order to have the CRA i n Theorem EU7 c a r r y through to the must be more concave utility function of the than i n the case where wealth u h(y). This is c r u c i a l f o r the proof of Theorem EU9. e at context of initial wealth not is because In the and Ug a r e lottery-specific, DRA, suppose an a g e n t ' s y is n . When y is - - 108 case with stochastic the more r i s k averse i n d i v i d u a l A that, A Then u ! , u! > 0 , A A deterministic. neither insurance Note w i l l be X and premium f o r i n c r e a s e d by a constant risk Ay i n every state, decreasingly chase same insurance premium change accordingly? risk we him to more r e l u c t a n t averse, insurance. policy holding sense how w i l l h i s of of Thus, should the premium he decrease. risky DRA i s the expect asset given in become will Similarly, when he gets be we prepared expect richer. Theorem EU10 which to him to If pay simply to is pur- for the increase his The c o n d i t i o n is he for this a rephrasing of Theorem EU9. Theorem EU10 (DRA w i t h continuous, y strictly Morgenstern u t i l i t y f u n c t i o n u(y) (Strong A r r o w - P r a t t (b) (Insurance (c) insurance premia and Q y^ = y +Ay, ple each of Demand) where E ( t i | y ) = E(r\) t^y^ IC is decreasing = it(y e) q o > stochastic Then, for a l l implies q safe Ay > 0 i m p l i e s levels TI > %^ f o r u's y Q all Q the following and x^ are u's asset - property: risky-asset respectively, r and r i s k y Then, Q 109 k. y . > 0 for a l l y . - Neumann- i n y for a l l i n i t i a l wealth Q with von of a and iz^ = it(y^,e) are and y^ = y + A y , Q properties equivalent: Suppose X levels y set-up u e at = E(e) the above i n i t i a l wealth o risk respectively. Q portfolio U The f o l l o w i n g twice-differentiable are Suppose for E(ejy ) (Risky-Asset at Index) - Premium) e satisfying A y , Ross): increasing, (a) In a d d i t i o n , and Q X Q < ^ if asset demands i n a simz = A y > 0. r+T), CRA and DRA with Stochastic Wealth Beyond Expected U t i l i t y 6.2 To WU, an derive the we apply the condition 2 risk Var(ejy ) = a and respectively. y +e, Q comparative same approach as infinitesimal o for for i n the £ contingent every y e Y. Q and d e c r e a s i n g GIPRA under preceding subsection. on y Consider e Y with E ( e | y ) Q Q = 0 and L e t F and G be the d i s t r i b u t i o n s of A WU a g e n t ' s insurance premium i s y Q d e f i n e d by (6.6) below: Jv( y - n) ( y - it) dF o w / o v Y (y + £ 0 = Jw(y -it)dF w £ ) (y + ) d G + 0 Y O * o» In e ) " view of speculate that is true Y 6 ' 6 ) cross m u l t i p l i c a t i o n y i e l d s ( ;F)dF ' s i m i l a r i t y between (6.7) the -C"(y+k>F)/C'(y;F) for. decreasing and (6.4), one i s i n y for This conjecture all 6 ' tempted k and F i s 7 ) to the turns out to be i n c o r r e c t . that C (y+k;F) q(y+k;F) C (y;F) C (y;F) A all ( 0 -/C'(y ~2 A for d F Q / C(y ;F)dF 2 c o n d i t i o n we are seeking It (y ) o Expanding both s i d e s v i a T a y o r ' s s e r i e s a f t e r y w 0 J w(y +E)dG+J;£w(y )dF o a v Jy (y ) B k and F i f and o n l y i f , > 0 and a d e c r e a s i n g , f o r every F E D , there e x i s t a constant concave f u n c t i o n h^(y) such that C (y;F) = ^ C ( y ; F ) + h ( y ) . A Since C and A result, B C are B p F-specific, X and h must also depend on F . As a the proof of Theorem EU9 w i l l not go through f o r WU. From (6.7) we know t h a t , must be t r u e if %^ i s to be g r e a t e r that - 110 - than TCg f o r a l l Y , it C (y;F) q(y;F) A /C (F)dF " A for all y, for small r i s k s , ( " /C (F)dF > 6 , 8 ) B where F is it the d i s t r i b u t i o n of y . turns out similar A l t h o u g h (6.8) is derived to Machina's c o n d i t i o n f o r general risks. To extend Machina's results from s t r u c t u r e on LOSUF C as below (Chew, Assumption 6.1: preference ante The l i n e a r i.e., there LGU, we impose additional 1985): Gateaux d e r i v a t i v e f u n c t i o n a l V ( » ) : D^ ^ R i s bounded, FDU to £(•;•): continuously J * Dj ^ R of differentiable e x i s t s M > 0 such that |c(x;F)| the and ex < M for a l l x e J and F e D j . In this follows, section, we the theorems suppose Assumption V., V with i n c r e a s i n g , t l o n 6.1 are and C satisfying Assump- i5 ( 6 ' 8 ) a l l y and F . (Insurance Then, Premium) \ addition, > \ if f o £ of r a 1 equivalent Suppose u A and %^ are the r e s p e c t i v e insurance 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 ^ s a t i s f y i n g E ( e j y ) = E ( £ ) at a l l y . 1 o b o t h V. and V_ are d i v e r s i f e r s , A (c) what two LGU f u n c t i o n - B premia f o r r i s k is C of " /C (F)dF > A In In q(y;F) A (b) satisfied. equivalent: " /C (F)dF for concave LOSUFs A C (y;F) ( a ) properties 15 A is are s t a t e d i n terms of LGU o n l y . Theorem LGU9 (CRA w i t h y ): The f o l l o w i n g als 6.1 then each of the above B to: ( 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 A - B Ill - and O , Xg a r e lio their respective demands i n a simple portfo- s e t - u p with safe a s s e t r and r i s k y a s s e t z = r+n, where E(n|y ) = E(TI) > 0 for a l l y . risky-asset Then, x. Theorem LGU10 functional are ( a ) y B and A y ) : Q V with i n c r e a s i n g , The f o l l o w i n g p r o p e r t i e s of an LGU concave LOSUF C s a t i s f y i n g Assumption 6.1 equivalent: - g"(y;F) ; _ > /C'(F)dF for (b) (DRA w i t h < x,,. A O o all y* e at Then, \ > r , . /C'(F*)dF* > y (Insurance risk C"(y*;F*) and F*(s) = F(s-A), A > 0. Premium) Suppose % and i n i t i a l wealth f \ o r all levels ^ satisfying are C's y Q i n s u r a n c e premia f o r and y^ = y + A y , respectively. Q E(e y ) Q = E(e) for a l l y Q if Ay > 0. In a d d i t i o n , i f V i s to the (c) a diversifier, then each of the above i s equivalent following: (Risky-Asset Demand) L e t X q and x^ be C's r e s p e c t i v e risky-asset demands at i n i t i a l wealth l e v e l s y^ and y^ = y +Ay i n a simple p o r t Q folio set-up E ( r i | y ) = E(TJ) o with equivalent: / x ... n a ) y and Q _ C"(y;F) /C(F)dF > r and r i s k y a s s e t Then, X Q V with i n c r e a s i n g , are ( asset > 0 for a l l y . Theorem LGU11 (DRA w i t h functional safe Q Ay): if A y > 0. concave LOSUF C s a t i s f y i n g Assumption 6.1 . ( /C*(F*)dF* i 1 where The f o l l o w i n g p r o p e r t i e s of an LGU C(y*;F*) f o r a l l y* > y and F * > < * z = r+rj, F. - 112 - 6 ' 1 0 ) (b) (Insurance risk e at E(e) In a d d i t i o n , equivalent if > levels y f \ Q for a l l y J (c) % Then, TI and it^ are Suppose i n i t i a l wealth > 0. Ay Premium) o r a 1 1 and y^ = y^+Ay, Q J o V is ^ satisfying E(ejy ) then the above o a diversifer, at portfolio Suppose X each of and x^ q i n i t i a l wealth levels o o conditions is C's respective are y and y, > 0 for a l l y o Q risky- = y +Ay > y and y +Ay. in a z = r+T), Then X Q Q r Several £ in points are insurance worth noting in the above premium c o n d i t i o n and the theorems. c o n d i t i o n are r e q u i r e d to be u n c o r r e l a t e d w i t h y manner that all E(e|y ) = E(£|y +Ay) o y o and y + A y . Q premium and risky-asset sources of risks tion to eliminate demand interact. Q and Ay i n such a o wealth will risky, naturally The purpose any p o s s i b l e is of offsetting the = E(TJ) > o an a g e n t ' s depend on insurance how different uncorrelatedness or the risky-asset = E ( E ) and E ( r ) | y ) = E ( T l | y + A y ) When h i s Q First, r i s k n i n the demand is = E(ejy +Ay) = s e t - u p w i t h safe a s s e t r and r i s k y a s s e t where E ( T ) j y ) = E ( r i | y + A y ) = E(TI) for and o Demand) demands simple 0 respectively, to: asset risk premia f o r and y +Ay. (Risky-Asset <x C ' s insurance aggravating restric- effect among risks. Secondly, Machina for comparative the Fre"chet statics result in Theorem dif ferentiable LGU11 was utility. u t i l i z i n g path d e r i v a t i v e . originally Essentially, proved by proof is F~ o is the For example, if ~ y indifferent to F ~ , then y -ir o o J there e x i s t s a path from F ~ to F ~ ~ a l o n g y -it y +E o o o J - 113 - which the an same u t i l i t y indifference Tversky an premium the insurer insurer e at risk or the insured the scenario insurance. e occurs, is to In other lower away with n:( a) i n c r e a s e s indifference the premium, in path to this path Kahneman an agent can than the complete r i s k i n the for leaving event is and purchase insurance l o t s w i l l be drawn to determine absorb the words, similar Suppose i n s u r e r w i l l be r e s p o n s i b l e walk the & a a premium it( a) the Naturally, Along maintained. Consider If will is probabilistic for With a chance, risk. curve. (1979)'s insurance level whether it occurs. e; w i t h 1-a chance, the agent to the absorb the a. a a {F =aF~ , +(l-a)F~ , , -} , ^ - V ( F ) y -it(a) y -it(a)+e ' d a ' o o N N J v J = 0 at all a indifference e [0,1]. Similarly, given {F* =aF~ ~ +(l-a)F~ , ~ ~} . y +Ay-n*(a) ' y +Ay-it*( a)+e v decreasingly a path r N L N v r i s k averse, then %(a) > n*(a) a JF =aF~ , ~ + ( l - a ) F ~ . . ~ , . ~} . y + A y - n;(a) y +Ay-n;( a)+e' N 1 for N at Since Q wealth y +Ay, a lower Q J is If another the agent & o Q optimal there a path o is Ay > 0 , each a e [0,1]. it( a) i s ' v a w i l l be p r e f e r r e d . Define too h i g h to be & Hence ^ a V(F ) < 0. When C satisfies generalized will smooth continue Recall Assumption paths (see that A the effect y and F . on Chew ( 1 9 8 5 ) ) . path As differentiable such, Machina's the CRA c o n d i t i o n we d e r i v e d f o r i n f i n i t e s i m a l /C (F)dF all V is on all results to h o l d . q(y;F) for 6.1, the risks is: C (y;F) B > ~ /C (F)dF trx /t.\ji7 In the ratio B (6.8) context caused of DRA, we must by the - 114 shift - of take into consideration distribution. After a deterministic increase i n ex post wealth Ay, the d i s t r i b u t i o n of the wealth w i l l be F ~ . y+Ay. The c o n d i t i o n f o r DRA t h e r e f o r e becomes /C(F~ ">dF~ y+Ay y+Ay J g (6-11) ; d e c r e a s i n g i n both y and Ay. When ratio final the (6.11) first-degree wealth This i s increment decrease in stochastic y as is the case of Theorem LGU10. stochastic, well as dominance. in it is required distribution Hence, in condition that the (a) the sense of of Theorem LGU11. O b v i o u s l y (6.10) i m p l i e s consider a positive stochastic E( 0 y ) = 0 f o r a l l y o agent who wealth less is by a zero-mean therefore become 6 is risk averse is added more risk to averse. ambiguous sense that caused it in general. by A n o t be in the his wealth, The net Arrow. A, he by the o p p o s i t e will effect is the e f f e c t he of As will his become When an u n c o r r e feel an worse-off, uncorrelated increase i n wealth stronger than (6.9) on an a g e n t ' s r i s k effect A > 0, C o n s i d e r an of condition (6.9). C o n d i t i o n (6.10) offset sense amount deterministic f u r t h e r r e q u i r e s that Ay = A+9 w i t h from below by - A . deterministic i m p l i c a t i o n of risk the d i s t i n c t i o n between them, increase bounded zero-mean r i s k and a simultaneous is To see wealth positive, r i s k averse — the cted, and o decreasingly increases (6.9). however in the attitude of any zero-mean r i s k 9 bounded from below by - A . This stronger measure can be rephrased case where agents have i n d e n t i c a l s t o c h a s t i c case, agent V . i s s a i d to be more r i s k averse C (y;F) Cg(y;G) /C (F)dF /C (G)dG A A B - 115 - to characterize CRA f o r i n i t i a l wealth y . than agent V,, the In such a up to A if for all y, the d i s t r i b u t i o n of Also 9 and F , where note s+9 i f that there Neumann-Morgenstern u t i l i t y 9 satisfies F is is that of no E(9) = 0 and min{9} > -A and G i s s. Theorem EU11. f u n c t i o n u(y) This is because WU10 wealth and increment WU11 are c o u n t e r p a r t s without are omitted stochastic. because the d i v e r s i f i e r - On the they will other be requirement. 116 von does not depend on d i s t r i b u t i o n , r e n d e r i n g EU i n c a p a b l e of h a n d l i n g the s i t u a t i o n where both i n i t i a l and the - hand, identical wealth Theorems WU9, to t h e i r LGU 7 CONCLUSION After economics rections mary an extended and f i n a n c e , being impetus utility. behind is include prospect the to Meginniss (1977) (1979), validity construct theories the of excitement Descriptive the theory predominance a sense of many attempts Handa (1977), Tversky's there contemplated. They (1954), p e r i o d of of has p r o v i d e d the of Looms and Sugden (1983), M a c h i n a ' s Fre*chet d i f f e r e n t i a b l e tional analysis 1979a; 1979b; 1984). ries, Chew, In o r d e r to 1980; 1981; the pected utility 1983; Kahneman and Bell (1982) preference 1983; d i s c r i m i n a t e among these a l t e r n a t i v e s t u d i e s w i l l be needed Edwards and func- (Chew and MacCrimmon, Fishburn, way of d i s c r i m i n a t i n g among them i s economics of uncertainty utility, reported (1980) 1982; 1979), Nakamura, preference to d e l i n e a t e theo- their res- domains of e m p i r i c a l v a l i d i t y . Another to few date. such and i n f o r m a t i o n . applications Of the applied prospect behavior. of via their In comparison with alternative theories 'misperception-of-probability' theory to account applicability for several have theories, puzzles in S h e f r i n and Statman (1984) p a r t i a l l y a p p l i e d p r o s p e c t model i n v e s t o r s ' preference Among the t h e o r i e s both 1982b) and weighted further experimental pective to (1982a; pri- expected (1978), theory in beyond (1953; and Karmarkar utility i n terms of new d i - theories Allais regret expected weighted utility f o r cash d i v i d e n d s over s t o c k of g e n e r a l p r e f e r e n c e and Fre*chet - functionals, dif f erentiable 117 - exbeen Thaler consumer theory to dividends. implications preference of functional approach f o r Chew income (1983) weighted tible tion inequality and Machina (1982b). with the the conducted by applied 'discrepancies' first-price Cox, Fre*chet risky-asset Roberson demands. of its are and finance. theoretical lines of given theory tions that the soundness Machina, and to increase alternative appear model can bids reported (1982). in that the Machina theory to as w e l l as is compa- Dutch a u c experiments (1982a; obtain 1982b) conditions for the since the for n o r m a l i t y of and most scathing of homogeneous under the attack of M a c h i n a ' s and E p s t e i n ' s works, Statman, be for preference and the for papers utili- powerful by Borch (1969). f i n a n c i a l markets. tions derived and p r o v i d e d a r e f r e s h i n g in to auction aversion, fragmentary research observed Smith rather In order numerous the utility risk analysis With the e x c e p t i o n the (1982) respective E p s t e i n (1984) a p p l i e d F r d c h e t d i f f e r e n t i a b l e (1969) and F e l d s t e i n tigations and dif f erentiable to mean-variance defense in sealed-bid comparative and d e c r e a s i n g Shefrin Weber i n the u t i l i t y agents a Nash e q u i l i b r i u m b i d d i n g s t r a t e g y and ty were presented Among them, Epstein have our understanding preference theories worthwhile. specific compared to The other is risky-asset those to on d i r e c t e d towards relevant the is to to studies of to a p p l i c a b i l i t y of economics, directly and o b t a i n expected inves- relevance financial i n finance based the direct of The f i r s t situations properties n o r m a l i t y of nature. the above utility two apply a implicain the the d e r i v a t i o n of c o n d i - finance demands. This weighted utility such as essay i s risk intended aversion towards latter. We expected focus our utility attention and l i n e a r on Gateaux u t i l i t y - 118 - and which i s contrast Fre*chet it with differenti- able when restricted Gateaux u t i l i t y to separately bounded domain. are s e l e c t e d because, both are a n a l y t i c a l l y Under a expected Weighted u n l i k e other utility, if mixed with a t h i r d curves two lotteries lottery i n any simplex straight lines. compound lotteries curves. Intuitively, indifference at the ted), and w i l l (i.e. consistent convenience, involving To accommodate must curves proposed and linear alternatives, tractable. are the that behave with be most to lie i n conformance first-degree choice on two compromise intersect (or with the are distinct is to law of when indif- parallel these two indifference permit a set of w i l l be v i o l a - the-more-the-better dominance). indifference the behavior, transitivity stochastic we may a l s o r e q u i r e the then that 3-outcome l o t t e r i e s liberal do not This implies Allais-type allowed indifferent, 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 a l s o be i n d i f f e r e n t . ference utility curves For to be technical continuous and smooth. It is preference be not surprising It being is therefore referred to. mean p r e s e r v i n g any p r e f e r e n c e theory, tional certainty ving so little spread, structure spread are e q u i v a l e n t transitive, degree stochastic to among equivalent, on the problem c o n t e x t s might not specify imposed the others, sense risk of unconditional certainty and r i s k y - a s s e t demand. risk aversion Without in equiva- specifying we proved that r i s k a v e r s i o n i n the sense of equivalent plete, necessary We d e f i n e d , terms of c o n d i t i o n a l c e r t a i n t y lent, with ordering, r i s k aversion i n different equivalent. aversion that condi- and r i s k a v e r s i o n i n the sense of mean p r e s e r as continuous dominance. long as the underlying preferences in distribution, This - was first 119 - and c o n s i s t e n t showed for are com- with finite first- lotteries involving rational probabilities, ries. This risk if a l s o holds averse in the then extended comparative than agent B i n the sense of and o n l y i f A i s more r i s k averse sated spread. being more implies We a l s o risk that A will that, than B in demand l e s s context, i.e., conditional certainty regardless terms of the of of the simple risky asset lotte- agent A i s utility compen- theory, compensated in more equivalent than B i n the sense of simple showed averse to g e n e r a l monetary A spread a world with one s a f e a s s e t and one r i s k y a s s e t . In expected u t i l i t y , properties of a preference o r d e r i n g are c a p t u r e d i n the a g e n t ' s von Neumann-Morgenstern u t i l i t y we can i d e n t i f y tability of its non-expected a general utility u t i l i t y counterpart, functional will be and c a l l e d derivative LOSUF) nance C(x;F) which utility. at termed For i n s t a n c e , equivalent it the consistency equivalents £. a linear greatly role of can be used the global risk explain however, Gateaux utility why people Its GSteaux (abbrev. stochastic in and mean p r e s e r v i n g spreads to For von Neumann-Morgenstern aversion local risk aversion. trac- function- function with the f i r s t - d e g r e e U n l i k e expected u t i l i t y , to p o i n t w i s e on u t i l i t y utility If enhanced. Gateaux u t i l i t y . a lottery-specific an i n c r e a s i n g certainty by a concave cause is a functional i n many ways p l a y s requires tional not such function u(x). the a n a l y t i c a l 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 als largely the terms is condi- characterized concavity T h i s gap i s purchase of domi- insurance of C is welcome beand gamble the same t i m e . If linear expected u t i l i t y , a special can resolve why should we be i n t e r e s t e d case of l i n e a r GSteaux u t i l i t y ? - 120 major controversies i n weighted under u t i l i t y which is At l e a s t three arguments can be - made i n r e s p o n s e . It is unclear First what In c o n t r a s t , sumptions about will ty conform be v a l i d . is the and weak weak is is substitution. axioms, embedded in a consequence axiomatic. linear of as analysis weighted This f o r expected appears to render weighted to generate (cf. all utility indifference lity f a r more e f f i c i e n t . utility the indifference curves indifference curves are p e r m i s s i b l e . curve Of c o u r s e , Nevertheless, It func- and this the point also means at noninter- In comparison, weighted only requires the knowledge which a l l that when the problem context the l e v e l of l i n e a r Gateaux u t i l i t y , ted u t i l i t y may appear Most a i n a simplex of 3- there of one indifference will exist importantly, utiarbi- curves paradoxes t h a t can be e x p l a i n e d by l i n e a r Gateaux u t i l i t y but not by weighted ty. utili- F i g u r e 1.1), the amount of i n f o r m a t i o n needed under secting intersect. maker's u t i l i t y when a n o n l i n e a r p r e f e r e n c e l i n e a r Gateaux u t i l i t y might prove insurmountable as any smooth, trary continui- a decision via as- for. outcome l o t t e r i e s is Ga*teaux specific transitivity, As long the i s not the o n l y axiom that departs from expected substitution. called Secondly, utility are namely completeness, these Note that n a t u r a l replacement tional to l i n e a r Gateaux u t i l i t y properties weighted preferences, monotonicity, preferences all, preference utility. ty, of utili- does not r e q u i r e g e n e r a l i t y the much g r e a t e r efficiency of at weigh- attractive. the specific functional form of weighted utility a l l o w s us to s o l v e e x p l i c i t l y o p t i m i z i n g problems such as p o r t f o l i o selec- tion, impli- i n t e r t e m p o r a l consumption d e c i s i o n , cations linear in this essay are o b t a i n a b l e Gateaux u t i l i t y . One i s the - etc. For i n s t a n c e , under weighted observation 121 - some u t i l i t y but not under that, no matter how r i s k averse he may counterpart, be, a will weighted invest utility a stictly agent, positive like amount his in expected the l o n g as the expected r a t e of r e t u r n on the r i s k y a s s e t i s than the safe r a t e of r e t u r n . the sense of sets are than in his mean p r e s e r v i n g always risky-asset Also, spread) convex so demands. we are assured weighted that his Under weighted linear r i s k y asset strictly that utility agent's is as greater a r i s k averse utility Gateaux utility (in not-worsequasiconcave utility, this need be assumed. Another r e s u l t cient unique to weighted utility agent's then utility, weighted this utility performing condition to yield conventional is the obtained first comparative to l i n e a r Gateaux u t i l i t y without al Even form. linear Gateaux not o b t a i n a b l e The above search. tions though It problems utility without such via path be point out interesting intertemporal to is that obtained bid are sensitive to agents' under expected u t i l i t y auction are isomorphic, see be choice, altered approach is explicit not function- c a r r i e d out under solutions the - 122 further financial are and the it is and the and It hypothe- well first-price English auction - economic utility. expected u t i l i t y re- implica- information value For example, the Dutch a u c t i o n are for under weighted under the preferences. so can be the conditions, how market b e h a v i o r a l consumption will results statics i n some s p e c i f i c bidding strategy the optimizing assuming a s p e c i f i c a natural direction competitive possible Theorem WU8). order This suffi- structures. under expected u t i l i t y as and second differentiation, imposing more discussions should obtained some comparative by f i r s t statics. applicable that the n e c e s s a r y and c o n d i t i o n f o r the r i s k y a s s e t to be a normal good ( c f . With weighted sis is known sealed- second- price sealed-bid trate that they might maximizers price auction. with auction utility? sealed-bid not Will Weber (1982) be p e r c e i v e d decreasing, sealed-bid weighted Nonetheless, concave remain the auction continue as was isomorphic weight to demons- by weighted utility functions. isomorphic to demand-revealing the able Will English p r o p e r t y of to h o l d under weighted the second- auction the under second-price utility? On the other hand, some r e s u l t s o b t a i n e d under expected u t i l i t y prove robust s i o n i n the of to p r e f e r e n c e sense of mean p r e s e r v i n g conditional example. extent The utility from likely, and such in the sufficient an example. he it that the equivalence and the normality is facing has much to do with in weighted condition utility obtained for condition this the essay, will under might the affect his that, market c h o i c e between d e c r e a s i n g of risky-asset linearity of the call of such for an some utility. risky-asset the a t t r i b u t e s behavior. demand under sense when an a g e n t ' s concavity preference is expected normality reveals i n the aver- of of We l e a r n the utility expected utility f u n c t i o n a l — an a n a - state-independence. 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(Little, 1957), and the size of Examples i n adaptation range cipants loans of bulk of the credit is credit of from local to was not fixed, to are a u c t i o n e d who have not yet off received to loans While 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 prevalent among the developing developed c a p i t a l markets), funds, 131 and socio-economic in Keta, - and Ghana Chinese and the parti- 1962). with an e x p l i c i t (given a seniority loan i n t e r e s t - f r e e (Geertz, the exhibits those popular among the are other - The a c - the h i g h e s t b i d d e r among the countries there satisfy deposit, an a s s o c i a t i o n to association, i n which the o r d e r of r o t a t i o n was determined by deposit type, referred available size to and Japanese where the o r g a n i z e r gets the f i r s t subsequent sa- i n a r o t a t i n g manner. including recruitment p o l i c y , degree of v a r i a t i o n s and frequently method of d e t e r m i n i n g the o r d e r by which members r e c e i v e remarkable credit rotating In a r o t a t i n g a pool to capital small The forms of its most the mutual p r o v i s i o n of institutions deposits of relatively developing anthropological credit a developing consumers. the financial of loans a r i s i n g arrangements is members makes and o r d i n a r y vings the the their finance focus relatively commonly observed are under- non-market, expectation-based credit component. obvious rotating Neighborliness, Chinese association types t h e i r methods and of of the is determined association as rotating the versions, r a t h e r than some form of tors gift well carry a exchange, significant poker credit members is rotating- clubs, etc. several variations and g i f t association are for funds or other mainly by the submitting since its 'sociological' rota- criteria cre- found i n the Middle E a s t i s mainly f o r the purpose of pur- (among L e b a n e s e ) . automobiles The l a s t 1957). In in The r o t a t i n g d u r a b l e goods such as (Little, the exchange. differ not v e r y i n t e r e s t i n g rates of r o t a t i o n among members. compete by s e n i o r i t y interest of neighborliness o r d e r of The A f r i c a n v e r s i o n order chasing as determining Japanese sealed-bids. dit that summarizes the c h a r a c t e r i s t i c s credit various tional activities examples. T a b l e 0.1 The exchange (among I s r a e l i s ) example of the rotating and r e f r i g e r a credit associa- tion refers to the b i l a t e r a l p r i v a t e arrangement popular among some I n d i a n laborers.* Under t h i s arrangement, two i n d i v i d u a l s at f i x e d In A s i a n intervals, agriculture-based accomplish their seasonal pioneering days, similar fighting fires, often amount harvest in alternated neighbors rotation. arrangements is between on pay days. communities, were In common gather America, for efforts during building to the houses, etc. The r o t a t i o n a l nature i n h e r e n t i n the context of wedding g i f t s . These examples a fixed in gift exchange i s p a r t i c u l a r l y clear Most people begin by saving were p r o v i d e d by the a u t h o r ' s - 132 - (giving c o l l e a g u e s and f r i e n d s . wed- Table 0.1: Examples of Rotational Exchange Examples Rotating Credit Associations C h a r a c t e r i s t i c s Chinese Japanese African Middle East Indian Laborer Neighborliness G i f t Exchange tradition tradition Organization nature explicit organizer purpose a bilateral private arrangement an informal f i n a n c i a l market no yes s a t i s f y i n g borrowing or lending needs no mutual aids friendship, etc. self-selection mutual s e l e c t i o n Membership qualification relationship Information (creditworthiness based on) mutual s e l e c t i o n multiple b i l a t e r a l m u l t i l ii t e r a l bilateral k i n s h i p , job immobility, e t c . locatlonal immobility status immobility Deposit form sice regular c o n t r i b u t i o n ($ or kind) fixed or variable frequency a i d to neighbors g i f t s given out variable fixed fixed variable Withdrawal form size frequency loan ( p a r t i a l l y refund, $ or kind) f i x e d or variable g i f t s received a i d received variable fixed f i x e d and once per cycle Assignment mechanism precision of rotation by competitive bidding by s e n i ority precise by l o t alternating by needs by t r a d i t i o n e.g. weddings imprecise ding gifts) ings' towards and take marriages their on a l o a n of friends is at least credit the is of with, that are p a i d back over course From we may i n f e r the without where the see the insurance membership when a p p l y i n g f o r On fixed the the credit occurrence without moral sense that and by paying assignment mechanism rotating is withdrawal insurance the key general subsequent most- i n a wedding club a poker c l u b which organizer. a special form of The purpose the of the orga- the ar- the members and p r o f i t making f o r company an individual is For i n s t a n c e , (assuming side, Inc. that associated an automobile i n the U n i t e d S t a t e s people difference First of exogenous presumably hazards. depend on of Consequently, a member may never the insurance tell get policy premium p e r i o d i c a l l y . association. hazards do occur and withdrawals in 'sav- the truth insurance). deposit deposits the company a c t s as claims. information. be a n o n - d r i n k e r and non-smoker is as insurance owner i n s u r e d by P r e f e r r e d R i s k Mutual Insurance must their 'wedding c l u b ' c o n s i s t i n g insurance sharing for some p r i v a t e 'withdraw' an e x p l i c i t and p r o c e s s i n g risk they Of s i m i l a r nature can be made to policies organizer. when Interestingly, functions association by s e l l i n g days members of p a r t i a l l y exogenous. Some arguments rangement gifts and r e l a t i v e s . serves c e r t a i n s o c i a l nizer of of other e l i g i b l e ly rotating wedding all, from the the rotation withdrawals. are g r a n t e d , actual losses determined. - 134 - which recognized is is is financial prompted by random i n imprecise Secondly, even i f cases in again are the insured the s i z e of withdrawals which make The withdrawal the withdrawal events holders will randomly Even credit though the association repeated is anthropological about a c e n t u r y ' s intertemporal competitive j e c t of a r i g o r o u s microeconomic intertemporal b i d d i n g process dit association. ods, is the Because of results i n the the in the resource Milgrom, 1979; F o r s y t h e and I s a a c , Samuelson, auctions competitive of the associated 1981; (1979) Cassady (1967) studies were Wilson, 1980; Cox, 1979; Roberson and is and concerned source with of four the s e c o n d - p r i c e sealed bid a u c t i o n . English auction, The purchaser In c o n t r a s t , who stops the Smith, prices prices pays across types forms of a u c t i o n mar- Green and L a f f o n t , and Cox, Titus, (1980) of Smith for R i l e y and and extensive historical Walker, auction surveys. examples.) market forms that purchases nobody - the Many - the s e a l e d b i d a u c t i o n and is willing i n the Dutch a u c t i o n move downward. 135 1980; a l s o r e f e r r e d to S t a r k and first-price price downward p r i c e movement Ne- literature move upwards i n p r o g r e s s i v e l y the - time. have been numerous Smith 1982; anecdotal the peri- association H a r r i s and R a v i v , 1981; (The reader i s the Dutch a u c t i o n , the there various Coppinger, Engelbrecht-Wiggans a good cre- results. of Myerson, 1981; on the rotating bidding Oren and Rothkoph, 1975; English auction, intervals. (single-period) role sub- between b i d s a c r o s s independently of our allocation Milgrom and Weber, 1982). Rothkopf over. the rotating t h i s essay i s Chinese v e r s i o n the p o i n e e r i n g work of V i c k r e y (1961), of In The focus of the interdependency k e t s (Oren and W i l l i a m s , 1975; 1982; 1962), the b i d d i n g process has never been the be of h e l p i n the development Since 1977; old (Geertz, study. from having repeated vertheless, studies concerning I n t e r t e m p o r a l b i d d i n g process f o r the r o t a t i n g c r e d i t distinct will literature object at smaller to bid The b i d d e r that price. V i c k r e y argued that sealed the E n g l i s h a u c t i o n i s b i d a u c t i o n where rejected bid. This and Smith (1982). morphic to the is b i d d e r pays to the the second-price p r i c e of the highest supported i n the e x p e r i m e n t a l work of Cox, Roberson first-price sealed conjecture bid - auction own b i d ) - however, is the Dutch a u c t i o n i s (where the falsified highest i n the iso- bidder same e x p e r i - study. The o r g a n i z a t i o n of detailed dit highest V i c k r e y ' s other pays the p r i c e of h i s mental the isomorphic d e s c r i p t i o n of association, useful the r e s t the called Section thetical tigate 2 contains Hui (with this s t r u c t u r e of 'Hui', terms and n o t a t i o n s , of 2 is system given and d e s c r i b e in Section the 1 where we some p r e l i m i n a r y a n a l y s e s of 2 or 3 members o n l y ) . same time A more cre- introduce e i g h t a c t u a l cases of H u i . familiarize several small, The main o b j e c t i v e is to hypoinves- f o r the e x i s t e n c e of an i n f o r m 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 , and at o u t l i n e d below. the Chinese v e r s i o n r o t a t i n g i n a p r e l i m i n a r y way the r a t i o n a l e financial essay i s the western-derived reader with the banking workings of Hui. Section comes, presents preferences agent's with, 3 and intertemporal but does not The l a b e l the assumptions expectations. reservation depend o n , 'Hui' is model's discount agents' used i n both the - We a l s o 136 on an state vector, having access individual's a definition which to an compatible some i n t e r e s t s i n g u l a r and the p l u r a l - is of in- form. rate in a This formal allows strategy financial us to under second-price decreasing on h i s also derive, the on marginal subjective discuss with his winning obtain we, in Section This tion. each sult. to bid, a tacit sible rates. including rate (and the restriction (increasing statics uniqueness. i n d i v i d u a l optimal auction additional tractable 5, of bidding hypothetical concavity utility function, marginal outbidding and e f f i c i e n c y and and implications a rate) each p e r i o d . impose for each form f o r We of the further agent, a Nash e q u i l i b r i u m b i d d i n g restrictions, his ex post including the one for the period the ante among a s m a l l to a s i n g l e (nondeterministic) an a p p l i c a t i o n of buyer group of (e.g. the suppliers federal - 137 - the rates in re- i n Section 5 selling an government). directions auc- winning interest the model b u i l t t h i s essay by s u g g e s t i n g some p o t e n t i a l research. realized i n which he wins N a s h - e q u i l i b r i u m - i n d u c e d p r o b a b i l i t y of ex risk Hui borrowing and These r a t e s depend on the h i s t o r y of 6 describes collusion commodity 7 concludes a corresponding Section the and strategy. yields, Weighted by the period, existence Neumann-Morgenstern some comparative lending interest its 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 to neutrality. 4, first-price outbidden order strategy, prove Section the von i n d i v i d u a l optimal bidding In and in observed auction) time-additivity market, for indiviSection further 1 THE GENERAL STRUCTURE AND ACTUAL CASES OF HUI 1.1 The General Structure of Hui A Hui c o n s i s t s whom he b r i n g s rowing fault NA, of together and l e n d i n g which shared is repaid For operating by the N members who bid succeeds submitted count-bid' actual installments on the of A at other collectively. hand, each of poses Consequently, replaced e f f e c t i v e l y organizer and n (= 1, a their the bor- the de- l o a n of N subse- common risk an otherwise m u l t i - by b i l a t e r a l 2,..., choice ones. N) denote the p a r t i c i - the n t h p e r i o d . Let b ^ by p a r t i c i p a n t i at p e r i o d n , and b^ = max{b^ } be n at before-discount member's satisfy market^ and b e a r i n g i n b i d d i n g f o r the p o o l at h i g h e s t b i d submitted period, organizer, exchange r e l a t i o n i s L e t 0 denote the the of h i s the o r g a n i z e r r e c e i v e s an i n t e r e s t - f r e e i n N equal The and N v o l u n t a r y members form an i n f o r m a l market to needs. periods. lateral the to r i s k of each member, quent pant an o r g a n i z e r period n. (or payment We denote by A the before-premium) deposit at is each period The s e r v i c e s p r o v i d e d by and s e l e c t i o n of members, and d e l i v e r y of d e p o s i t s . the o r g a n i z e r the e x e c u t i o n - 138 - into of the be the per- the pool. The to A . In a 'dis- funds pays A at every related H u i , each member who has a l r e a d y r e c e i v e d 'size' n include competitive recruitment of a u c t i o n s , and the c o l l e c t i o n subsequent p e r i o d . A-b^ apiece. before At p e r i o d n , In a those who have not yet b^, he has to pay A+b at n Once he wins, every p u l a r among the Chinese w h i l e the latter king. flow for the patterns Although the J a p a n e s e ' s l i - and the N members i n both a d i s c o u n t - b i d Hui and a premium-bid Hui are summarized i n Table From Table 1.1, Hui is also the the N members (N-l)b^ in receives NA at at in the a must be every first bidding. ers and periods p e r i o d N, i s granted his one leaving t h i s Hui i s Hui, is status participants, to last. 1, who Note wins N, i n a that, the member, last 4 A Hui formed among pool NA- a pure borrower, whereas member N, who The other members changes to a borrower. Since the fund a v a i l a b l e the number of member O b v i o u s l y , an a t t r a c t i v e lenders. member 1.1. lie a member remains a l e n d e r u n t i l he to b i d . the the number of a pure l e n d e r . In g e n e r a l , to that bidding, are e l i g i b l e period, clear discount-bid which p o i n t only lenders is number of where i n between. loans, it period the former i s more po- seems more to organizer pay say at p e r i o d n subsequent p e r i o d . both d i s c o u n t - b i d and premium-bid Hui are o b s e r v e d , The cash loans ' p r e m i u m - b i d ' H u i , each member pays A at every he succeeds i n b i d d i n g f o r the p o o l . by b i d d i n g obtained to bidders collect At each receives period, at each p e r i o d decreases NA at some- the end by one without Hui c o n s i s t s of a 'good' mix of borrow- by a homogeneous group of borrowers will When no bids are submitted ( o r , e q u i v a l e n t l y , a l l bids are z e r o ) , the winner i s determined by l o t . In the case of t i e - b i d s , e i t h e r the fund is shared e q u a l l y (consequently, f u t u r e repayments are a l s o shared e q u a l l y ) , or a second-stage b i d d i n g i s conducted to s e l e c t the winner. - 139 - T a b l e 1.1: Cash Flow P a t t e r n s a. period of Hui P a r t i c i p a n t s Discount-bid Hui 0 1 ... n 0 NA -A ... -A ... -A -A 1 -A A+(N-l)(A-b ) . . . -A ... -A -A • • -A -A • • N-l N participant 1 • n • -(A-b ) -A ... 1 • nA+(N-n)(A-b ) n • N-l -A "(A-b^ -(A-b ) N -A "(A-b^ -(A-b ) 0 1 0 NA -A 1 -A NA • • ... n (N-l)A+(A-b _^) - A N -(A-b^) n b. period ... NA Premium-bid H u i n ... N-l —A ... -A N participant n -A • — A ... •• • -A -(A+b^ -(A+b^ -(A+ ) • • • NA+zJl]^ -<A+b ) -CA+b ) • • • • n N-l -A -A ... — A ••• NA+Z^b N -A -A .•• -A ... -A - 140 - Dl i n "(A+Vi) N A + E b i=l i provide l i t t l e room f o r t r a d e s . Similarly, H u i to be q u i t e a boredom s i n c e no one w i l l g g e s t s a dual problem to An e f f e c t i v e jective choice grees variable of consists features this of the mix of except to such as the r o l e of d e f a u l t This a su- as h i s ob- from members' p a r t i c i p a t i o n . His membership i n terms de- We s h a l l not point find the o r g a n i z e r ' s problem. o r g a n i z e r presumably maximizes l e n d i n g needs. essay be b i d d i n g a c t i v e l y . t r e a t e d here — ' s u r p l u s ' generated borrowing or in one competitive) f u n c t i o n the problem 1.2 (i.e. the a group of l e n d e r s w i l l out study from time to of the time their organizer's his salient risk. Actual Cases of Hui In November 1983, largest-ever-scale town of were the f i n a n c i a l sector Hui d e f a u l t i n her h i s t o r y . 100,000 people named C h i a - L i . involved for a t o t a l i n Taiwan was s t a r t l e d amount of This by the occurred i n a small A l l e g e d l y , over one thousand people four b i l l i o n New Taiwan D o l l a r s (NT$) ( a p p r o x i m a t e l y US$100 m i l l i o n based on the c u r r e n t o f f i c i a l fixed exchange r a t e US$1 = NT$40). This incidence has led several legislators to urge for governmental r e g u l a t i o n on Hui o p e r a t i o n i n Taiwan and has prompted at l e a s t one on Hui s t a t i s t i c s . jects' reluctance ted that tion; Qualifying his as conservative due to sub- to r e v e a l t h e i r a c t u a l involvement, Wen L i Chung estima- the t o t a l Hui membership approximates 85% of the i s l a n d ' s popula- the c r e d i t p r o v i d e d by Hui i s US$2.85 figures survey b i l l i o n annually, income (Chao-Ming, which i s r o u g h l y US$237.5 m i l l i o n per month, or about 1983). - 141 - 21.92% of the island's national lect In o r d e r to familarize i n Table 1.2 e i g h t a c t u a l examples among the nistor employees who of co-signed formed among the the a CPA f i r m ; all other employees of starting its senior, tenured and ending membership N time, with its workings o r g a n i z e r was pay Hui 1 was formed the p e r s o n n e l admi- checks. very low t u r n - o v e r . the of H u i , we c o l - Hui 2 - 8 were state-owned Taiwan Power Company (known employees. (excluding the found i n Taiwan. employees' the as T a i Power) which experiences uniformly reader Each Hui i s predetermined the T h e i r o r g a n i z e r s were fixed organizer), its characterized payment A , the type by its size of (discount-bid or p r e m i u m - b i d ) , and the a c t u a l winning b i d s . Organizing According to the current practice will draw up and then c i r c u l a t e tion p r o p o s a l with proposed and frequency of meetings features as $901 such and $904 the bids among p o t e n t i a l size (e.g. of payment every minimum amount will be i n Taiwan, a prospective participants organizer a Hui forma- (A) and membership (N), date two weeks or every month) and other of bids, considered as rounding-off $900 and $905 policy (e.g., respectively), etc. All of interested terms. parties Based on the response may be r e v i s e d . ing the all members. and s u g g e s t i o n s , terms and the names U s u a l l y the form i s designed n i n g b i d and the amount of these forms stopped of members w i l l be d i s t r i b u t e d with space to f i l l Because 142 - to i n the w i n The data i n the i n d i v i d u a l s who p r o v i d e d r e c o r d i n g the winning b i d s - terms a form c o n t a i n - the r e s u l t i n g p o o l at each p e r i o d . are taken from such forms. alterations i n i t i a l l y proposed When terms and memberships are f i n a l i z e d , agreed-upon T a b l e 1.2 are i n v i t e d to s i g n up and suggest after they obtained funds, T a b l e 1.2: A c t u a l Cases of Hui Hui start end O c t . 75 J u l y 77 J u l y 77 Mar. June 77 Mar. 79 J a n . 1,000* N type min b 1,000 20 20 disc. disc. 79 J u l y 80 J a n . 81 J u l y 81 82 June 84 May 83 Mar. 79 F e b . 80 A p r . 2,000 2,000 18 disc. 23 prem. 2,000 30 prem. 5,000 winning 450 470 401 500 300** 120 140 150 155 150 135 125 110 110 100 115 125 140 130 100 50 110 * A l l Hui are on monthly 200 210 200 150 160 170 170 150 160 150 160 150 160 170 160 120 400 350 360 360 360 320 310 340 330 310 280 280 230 250 250 220 210 210 5,000 disc. disc. prem. 600 900 1,530 1,680 1,680 1,720 1,610 1,680 1,590 1,750 1,610 1,620 1,640 1,520 1,660 1,670 900 960 920 920 950 960 980 1,000 basis. * A l l amounts are i n New Taiwan D o l l a r s (NT$); US$1 « NT$40. * * The winner was determined by l o t due to absence of - 143 - 5,000 28 bids 460 500 600 680 600 550 530 500 560 500 500 490 510 600 500 430 450 400 350 400 400 420 450 83 47 300 time* 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 78 O c t . bids. 25 the data are incomplete. Although t h i s form i s informal (e.g., by the o r g a n i z e r and the members, e t c . ) cording convenience court r u l i n g , of the acceptable and l i a b i l i t i e s it and i s participants, as evidence i s not n o t a r i z e d , not signed p r o v i d e d m a i n l y f o r the it is, according to a re- recent f o r the e x i s t e n c e of f i n a n c i a l c l a i m s among Hui p a r t i c i p a n t s (Chao-Ming, 1983). Organizer In return auctions, ceeds to for the services i n c l u d i n g r e c r u i t i n g members, the winner, Hui. the and most i m p o r t a n t l y assuming the d e f a u l t organizer obtains an i n t e r e s t - f r e e Should any member d e f a u l t , the d e f a u l t i n g member's share and the H u i w i l l l o a n at o r g a n i z e r must ( i n the once w i l l sense that h i s transactions) n i z e r at the time a Hui i s individual circle zer is able where he i s or to depends formed. by the We have yet Such enforcement course cases such, were organizers either in a 'Informal cre- loss not position by the be w e a l t h y . to - obli- from a l l f u t u r e Hui either as a Hui o r g a n i - discipline c r i t i c a l l y on some g e n e r a l i z e d need the to p a r t i c i p a t e as a member) i n a i m m o b i l i t y of which formal c o l l a t e r a l i z a t i o n i s As take over to know a case i n which an known to have d e f a u l t e d before of start a n e c e s s a r y a t t r i b u t e of any Hui o r g a - o r g a n i z e a Hui (or a Hui member. dealings is the g a i n from d e f a u l t i n g from h i s be more than outweighed and other i n f o r m a l posed to be t r u s t w o r t h y and f i n a n c i a l l y capable of assuming the d e f a u l t i n g s h a r e s . ditworthiness' risk c o n t i n u e without i n t e r r u p t i o n . To be an a c c e p t a b l e o r g a n i z e r , one must be b e l i e v e d gations conducting 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 p r o - by a l l members, of his deter 144 - of continuous definition of an example. The o r g a n i z e r s defaults, or, if in our 8 default did occur, Hui had the ability 1 had access hood of the Power i s to a l l members' bership. the o r g a n i z e r ' s to a large F o r example, paychecks. is As to Hui 2 - 8 , rather slim According the organizer to our and/or many i n s t a n c e s on i t s because all of capital, would ability of observation, any members the orientation higher the job with T a i were of members, with higher from depends mem- formed mainly dispute. a Hui i n c r e a s e s businessmen. shares its they were v i r t u a l l y of level draw funds the r i s k of a and the mix of small and paying up t h e i r risk likeli- cases. Hui i n Table 1.2 are businessman bid the a By now, a l l the 8 Hui have ended without in their as than the p o o l i n most the s a v i n g purpose and i n v o l v e d no businessmen, risk. the o r g a n i z e r of c h a r a c t e r and c r e d i t w o r t h i n e s s , extent For instance, no d e f a u l t of loss. running away c o n s i d e r e d more v a l u a b l e depends for assume the organizers' Besides Hui to This because opportunity costs pool while the on the is if first subsequent success t h e i r b u s i n e s s or t h e i r a b i l i t y to borrow from other H u i . It is worth n o t i n g duced the s o - c a l l e d that the 'professional recent organizers' entrepreneurship i n a r b i t r a g i n g across tive This broad ventures. connections neighbors, tion etc., about full-time breed of friends, and consequently profitable involvement. reluctant his to It risk, accept Hui or c h a n n e l l i n g professionals is however an o c c u p a t i o n o f t e n a prudent people organizer whom he - 145 - is to have lucra- good and ex-colleagues, access to v a l u a b l e i n f o r m a - opportunities organizer funds usually colleagues, have s p e c i a l investment intro- who have p r o f i t e d from t h e i r relatives, Being a p r o f e s s i o n a l To minimize often with either worthiness. new boom of Hui i n Taiwan has not or p e o p l e ' s does not credit- necessitate taken up by housewives. must be selective f a m i l a r with as and is members. In immigrant Chinese communities i n North A m e r i c a , membership either the is often organizer, lity to where or be a saver make p e r i o d i c lottery tions, r e q u i r e d to it is is the i n the payments produce a guarantor i n i t i a l periods (Light, allocational a new immigrant 1972). mechanism f o r acceptable to e s t a b l i s h In A f r i c a and have e s t a b l i s h e d his to abi- and M i d d l e E a s t rotating credit a common p r a c t i c e t h a t new f a c e s are not allowed and have to be the l a s t seeking associa- to draw l o t s ones to withdraw funds u n t i l they are b e t t e r known t h e i r c r e d i t a b i l i t y ( A r d e n e r , 1964). Members Hui (Geertz, has reportedly 1962). 5 In a sense, titution reflects bile among the the presence Chinese for of informal f i n a n c i a l this a more f o r m a l aspect of the e x i s t i n g relationships (which existed are viewed among the members. Today, as more c o n s e r v a t i v e but most safer) joining people Hui that 'superior' not some have information uncommon f o r one are willing greater to involvement and by c a r e f u l to take realize a search 30% - 800 years ins- saving-oriented still Hui formed among d i r e c t l y or i n d i r e c t l y . risk with least and g e n e r a l l y immo- are i n d i v i d u a l s who know each other f a i r l y w e l l e i t h e r Naturally, at for higher businessmen. and other returns With measures, 40% a n n u a l i z e d rate of 6 by some it is return 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 a g r i c u l t u r e - b a s e d communities, the exchange commodity was o f t e n i n k i n d , being r i c e i n many c a s e s . Today, money i s the only known c u r r e n c y traded i n Hui p r e v a l e n t i n Taiwan. b For i n s t a n c e , p i c k a Hui whose o r g a n i z e r i s your next door neighbor who runs a TV shop and has enough TV s e t s i n stock which you can l a y your hand on i n time i f s i t u a t i o n c a l l s f o r such a u c t i o n s . - 146 - Frequency of Meeting ( B i d d i n g ) The frequency of Hui meeting the s i z e of cial Hui the payment ( A ) , the s i z e of background of such as meeting is It is the the members. eight to a l a r g e extent on the membership (N) and the Most s a v i n g - o r i e n t e d , cases we presented finan- middle-class-based are on monthly b a s i s . Bi-weekly a l s o q u i t e common. noticed Many Hui i n v o l v e d daily and b i d d i n g depends that speculative i n the Chai-Li Hui tend to have shorter s c a n d a l were a l l e g e d l y intervals. on a d a i l y or b i - basis. S i z e o f Membership The s i z e ship of given 47 the of membership i n T a b l e 1.2 is size uncommon of for a ranges from 18 to 47. saving-oriented, payment N T $ 5 , 0 0 0 . 7 monthly In g e n e r a l , the the s i z e of payment and the frequency of meeting are It is believed that the of default. Most people to two years so that is not It longer tend to a Hui l a s t s , prefer the p o o l NA i s ders. 7 Hui, especially s i z e of membership, interdependent. the g r e a t e r a d u r a t i o n of is its one and h a l f not too s m a l l and yet risk years the d e f a u l t risk unaffordable. is q u i t e common that more than one i n d i v i d u a l share one membership o r one i n d i v i d u a l assumes more than one s h a r e . ticular A member- interest as it Such p r a c t i c e s A college 1980. graduate's is tantamount allow Hui to starting to - permitting display monthly 147 - The l a t t e r coalition a somewhat salary was case i s greater about of par- among bid- degree of NT$12,000 in flexibility This i n accommodating more d e s i r a b l e seems among to a fixed parallel the menu of insurance policies market rather needs together w i t h the p r i c e he i s s i z e of saving where than and b o r r o w i n g . an i n d i v i d u a l specify the size chooses of his own w i l l i n g to pay ( R o t h s c h i l d & S t i g l i t z , 1976). A member i s outsiders before also allowed to sell his share the Hui ends as long as he i s from the o r g a n i z e r . able T h i s can a l s o be done i f t e e s the c r e d i t w o r t h i n e s s of the member(s) to other members or some to o b t a i n the a p p r o v a l the o r i g i n a l member guaran- he introduces. Discount-Bid v s . Premium-Bid The found bulk among ones. The Hui i n Chinese in Taiwan are the North A m e r i c a , on discount-bid the payment Hui. part This in greater that made by provides bidding. a member an who the participation. This is d i s c o u n t - b i d Hui are more of Hui. loan In the has contrary, for withdrawn winning a member premium-bid there are the with insti- obtains the Moreover, b i d s (it is likely a more to general the a member d e f a u l t s , defaulting structure is A in a active encourage impression 'exciting'. a d i s c o u n t - b i d Hui and A plus by funds take consistent f o r however premium-bid H u i . to is the o r g a n i z e r would be r e s p o n s i b l e below are Most Hui bid i n a discount-bid a discount-bid an o r g a n i z e r event not current incentive As a r e s u l t , The i n t e r e s t - f r e e types type. differences: premium-bid Hui and A minus b. of Although these two types are s i m i l a r i n substance, tutional a. of cannot per-period different, member's both amount being A i n winning b i d i n a i n a d i s c o u n t - b i d H u i are bounded from go - is the same (NA) i n beyond 148 - A), but may i n principle be very to large prefer partly c. i n a premium-bid H u i . discount-bid explain fact count-bid type. The p o o l available NA-(N-n)b wish those to in n of at who wish to a have society before the with most 'speculative' Hui. This has This reasons can at Hui are of least the dis- The former appeals win. The l a t t e r argument i s at to people who is least p r e f e r r e d by to some extent j o i n i n g a H u i , a p a r t i c i p a n t can s e l e c t a Hui that suit his wide-spread a d d i t i o n a l advantage participant's premium-bid ones. NA when they pay l e s s . since, therefore p e r i o d n i s NA+E^_^b_^ i n a premium-bid Hui and least the c h a r a c t e r i s t i c s In that a discount-bid o b t a i n at superficial d. the Hui over The o r g a n i z e r preference. illiteracy, of discount-bid requiring less f u t u r e payments are independent arrangements r e c o r d keeping as each of past winning b i d s . Bidding On the b i d d i n g day ( o f t e n d u r i n g lunch time of the pay d a y ) , w i s h i n g to b i d would submit t h e i r b i d s to the o r g a n i z e r . t i o n a l mechanism f o r Hui has tended members The l o a n a l l o c a - to be v i a f i r s t - p r i c e sealed-bid auc- tions . Until the recent boom of Hui i n Taiwan, vely 'calm' . For example, ther to A member who c o u l d not bid. Recently, After a. to simply behalf has become relati- d i d not show up f o r the b i d d i n g o f t e n more competitive. bo- autho- or informed the o r g a n i z e r of the b i d d i n g , only the winning b i d was Hui b i d d i n g phenomena a t t e s t b i d d i n g had been many saving-minded members r i z e d another member to b i d on h i s b i d i n advance. the his revealed. The following it: The i n f o r m a t i o n about needs f o r loans - 149 - is guarded as top p e r s o n a l secret to prevent b. Eligible signals bid) c. competition. bidders not (usually by in need talking funds casually their their bids would about organizers until the of identity the implications announce bidders for the all of the the amount give he false intends b i d d i n g meeting l e a k the submitted (except role strategically to bidders. f r i e n d s or the o r g a n i z e r cannot Increasingly, have of to induce h i g h e r b i d s from r i v a l Members would w i t h h o l d that d. strategic the to ensure information. b i d s without winner). This revealing practice i n f o r m a t i o n and l e a r n i n g may i n the Hui setting. It should however be noted those r e p o r t e d i n Table endship needs and s o c i a l funds eligible a to lar) . norms, This loan to his agree ailing parent, upon a low, to less-commercialized play a r o l e . him ( p r o v i d e d corresponds in some non-market economic still hospitalize bidders w i l l subsidized 1.2, that, the of factors, For i n s t a n c e , it is Hui such e.g. if fri- a member very l i k e l y that nominal b i d and e f f e c t i v e l y course insurance that he f u n c t i o n of is not as too all grant unpopu- a Hui and other in- f o r m a l market mechanisms. Bids What to factors increase mium-bids affect i n A and N as are expected l o a n i n the former i s The economic investment the to of bids? suggested by the be h i g h e r Intuitively, we expect data i n Table 1.2. than d i s c o u n t - b i d s since Also, the bids pre- size of larger. determinants opportunity non-investment level cost of needs f o r f u n d s , of a Hui member's b i d i n c l u d e at capital, his and a s t r a t e g i c - 150 - stochastic component or least his nonstochastic inherent i n most game b e h a v i o r . First, consider c i a l markets. al' if it enforce Hui is have investment essay, ' i n f o r m a l ' i n making use banking system. relevant other investment capital of the of formal the same f o r most in the perfection of i n g does e x i s t . est and rate, available, of but loan for the scale. the Loans process to f u l f i l l . is two is contrast, information generated (i.e. market enforcement is the bank conven- Therefore who do not interest rate many s m a l l for small c o u l d be the opportunities, bank l e n d i n g First of all, businesses and forbiddingly costly For example, effectively businessmen for the c o s t of interest rate however more c o m p l i c a t e d due to the i m - or more noncorporate loan i s to In Hui p a r t i c i p a n t s be f o r m a l f i n a n c i a l market. f o r an u n o b t a i n a b l e that evidence capital ration- Banks, which cannot demand more than the r e g u l a t e d ments d i f f i c u l t ing 'form- p r e f e r to d e a l w i t h l a r g e c o r p o r a t i o n s due to r i s k economy business considered individuals. The r e a l i t y i s the finan- can save with banks. would formal f i n a n c i a l market a p p l i c a b l e to him. formal non-default financial capital opportunities is the system). creditworthiness to enforce familiar cost legal F o r those people who have other investment capital in and c o l l a t e r a l i z e d Almost everybody opportunity s a v i n g which i s (e.g. institution The most of recognized from without (informal) cost a financial institution on e x t e r n a l l y non-default within). tional the In t h i s relies w i t h i n the from the Infinity. offer a rate - - 151 consideration consumption and the It small The i n t e r e s t is as h i g h as are require- a standard requirement f o r guarantors. inter- rate then not s u r p r i s 50% a n n u a l l y for loans from H u i . For people who p a r t i c i p a t e practice), the i n t e r e s t t y c o s t of capital. i n more than one Hui (which r a t e of other Hui might be the This is especially true for is a common relevant opportuni- the p r o f e s s i o n a l Hui a r - bitragers. It should however be noted such as Hui to s u s t a i n be e v e n t u a l l y real for an i n f o r m a l f i n a n c i a l over a long p e r i o d of time, channelled returns. that the pooled c a p i t a l must to e c o n o m i c a l l y p r o d u c t i v e a c t i v i t i e s In other words, 'rate return' funds c i r c u l a t i n g w i t h i n a Hui system g i v i n g h i g h r e t u r n s and to the would e v e n t u a l l y Chia-Li Hui fiasco, involved is participant Hui-with-Hui real lead to economic production a pyramid. i n which the most a widespread draws funds operation This sector, is of then evident the this in light s t r i k i n g and d e v a s t a t i n g practice dubbed Hui sup- If flow outside on Hui must be in never activities y by growth keep economic of which y i e l d ported the real the mechanism system. Hui system of the feature i n another H u i . was failure blamed C o r p o r a t i o n s i n Singapore d u r i n g 1972 for and 1973 of the 'feed-Hui-with-Hui', i . e . , from a Hui to make payment also 1983 the (Chua, of Chit a FeedFund 1981). s I r o n i c a l l y , the p o p u l a r i t y 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 i n a n c e through Hui a s i g n a l of h i s c r e d i t a b i l i t y . How can a person who cannot be accepted i n t o a Hui expect someone e l s e to guarantee h i s loan? But i f one can borrow from H u i , why should he need to borrow from the bank? y The ' r a t e of r e t u r n ' from Hui suggests a p o s s i b l e model which i s beyond the scope of t h i s e s s a y . - 152 - general equilibrium As funds, to the stochastic the s i n g l e chase or non-stochastic most important use of loans and r e n o v a t i o n . i 0 Purchases of non-investment needs for from Hui has been house d u r a b l e goods, children's pur- education expenses, wedding expenses and f o r e i g n t r a v e l are other uses of funds from Hui. It seems r e a s o n a b l e to say that Hui p a r t i c i p a n t s have a 'reservation price' f o r loans a v a i l a b l e at each p e r i o d l a r g e l y determined by h i s oppor- tunity cost elsewhere, (b) financial tent will capital his the which depends on (a) his investment and (c) way his p e r s o n a l needs f o r f u n d s . Hui p a r t i c i p a n t s calculate the the bank s a v i n g interest he can a f f o r d to g i v e up. added to the b a s i c ted a c c o r d i n g l y . saving Later v a t i o n p r i c e f o r loans Would How h i s to calculate those in the average as the bid from h i s we w i l l 'reservation his reservation the For i n s t a n c e , of their s a v i n g w i t h banks the h i g h e s t discount a premium w i l l formally define discount? discount bids the consis- be calcula- this reser- discount'. reservation saving-orientation, Hui. bound r a t e and the maximum a f f o r d a b l e b i d i s S e c t i o n 4 where we study the o p t i m a l b i d d i n g s t r a t e g y Due to is I f he has other use of funds, i n S e c t i o n 3, a Hui member bid deviates rate This is upper U s u a l l y , an I n d i v i d u a l whose o n l y a l t e r n a t i v e use opportunity c o s t of c a p i t a l i n the formal f i n a n c i a l market or other sources, with bids. of Not will in general. be c o n s i d e r e d in f o r Hui members. i n Table 1.2 i n t e r n a l rate of are lower r e t u r n of than the The r e q u i r e d minimum downpayment i s o f t e n 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 o b t a i n enough funds f o r t h i s purpose. - 153 - first winner of Hui 6 i s uncommon to have 30% - approximately 1.33% 50% a n n u a l i z e d ex post ( t o be d e f i n e d i n S e c t i o n 5, nessmen are Whether the involved. to be Note that example, large ket, ket latter default leads is Hui borrowing i n t e r e s t not rate i n many Hui where s m a l l b u s i - course any two Hui on Table but took p l a c e extent risk of it to have higher higher bids default or the risk. other way investigated. Hui 1 and 2 had the b i d type, a not D e f i n i t i o n 5.1) The perceived around has yet per month, whereas by i n t e r e s t time. Even i f segmentation might s t i l l are s t r i c t l y comparable. same A and N and were both of two years a p a r t . determined which v a r y over 1.2 As d i s c u s s e d rates i n the the above, formal they had e x i s t e d For discount- b i d s are f i n a n c i a l mar- c o n t e m p o r a r i l y , mar- prevent b i d s from a t t a i n i n g p a r i t y . Even though Hui 3 was s m a l l e r than Hui 2 by two members, we would pect the winning The f a c t cause of bids i n Hui 3 to that the b i d s the existence a p p r o x i m a t e l y double those i n Hui 3 were much lower than expected of to the organizer. keep and the fewer members a Hui has, each member, everything else being The o r g a n i z e r the more c o s t l y constant. of ex- Hui 2. c o u l d be beexpensive to the o r g a n i z e r i s to We w i l l is show i n S e c t i o n 2 that a Hui with o n l y two members can not a f f o r d to have an o r g a n i z e r . If all members' constant over time, creasing over time. gible none bidders of could be the who expectations one would expect This is opportunity their bids m a i n l y due to result n b i d streams of i n Table bidders' - 154 - on the rates average are monotone expectations remain to be i n - number of enjoy the d i s c o u n t . 1.2 changing rates. interest the d e c r e a s i n g ( w i t h payment A - b ) w i l l winning the and over or eli- Although time, it opportunity Learning As long as a member knows or i s of capital, spread between h i s issue. have deriving The d i s c u s s i o n servation discount of is and h i s straightforward. bid is seeking ment defense vals' bidders' bidding strategies, interest where here is: etc. opportunity cost are v a l u a b l e Do p a r t i c i p a n t s learning the a realistic learning issue First competition loan. in evident that 1979; bidding Myerson, model 1981; must useful In repeated for, in- single- say l e a r n about from past capi- information. abstract compete bidders or b i d d i n g s t r a t e g i e s of of H u i , we f e e l , is all, not govern- their r i - biddings (Green Milgrom and Weber, 1982). allow he critical To keep our current as on the for bidders' of still Bayesian i n his analysis simple bids not do in depend on general. - 155 dual Secondly, role for costly this - reasons, that single-period auc- a b i d d i n g , he drops out the Hui d u r a t i o n . gains following i n repeated once a Hui member wins f o r the r e s t bidding, as based T h i s suggests a weaker i n c e n t i v e bidders' bidders of re- behavior. context loses is Milgrom, In the tions. same group of it prices and L a f f o n t , 1977; Therefore, the projects, reservation strategic i n f o r m a t i o n on r i v a l s ' f o r m a t i o n from the d i s t r i b u t i o n of past winning bids? p e r i o d -auctions a more The under B i d d i n g t e l l s us that Hui p a r t i c i p a n t s do be- O b v i o u s l y , other risk attitudes, to estimate h i s o p p o r t u n i t y cost discount including actively discounts. A question reservation reservation strategically, tal, the able essay, past as the when a Hui b i d d e r a 'seller' of the information search. we w i l l winning assume that Hui bids or bids in Default Roughly s p e a k i n g , the organizer, are well-planned. recruit the At the nonexistent names other by The p l o t as many members as few n o n e x i s t e n t zer. there are two types of Hui d e f a u l t , is names. next the usually possible At the s t a r t few p e r i o d s , members exhausted, member. and he Most goes as organizer-caused follows. and add to the defaults The o r g a n i z e r will list of p a r t i c i p a n t s a of H u i , he c o l l e c t s NA as the o r g a n i - he submits h i g h b i d s obtains one i n c u r r e d by loans. After simply d i s a p p e a r s . his i n the name of list of those nonexistent A Hui member can a v o i d this type of d e f a u l t by i n s i s t i n g on knowing a l l other members. The other type of d e f a u l t able to from the make their pool, elsewhere. possibly Although sumes a l l d e f a u l t does not shares the of due results from one or more members' not being contributions to u n f a v o r a b l e outcomes organizer is ignore this type they have drawn funds of their investments i n p r i n c i p l e the r i s k s posed by members, completely after of it is a fact default risk o n l y one who as- that a p a r t i c i p a n t when he makes the d e c i s i o n to j o i n a H u i . While we r e c o g n i z e ting, we w i l l not i n t h i s Instead, it will the presence of the default risk i n the Hui s e t - essay attempt a model f o r m a l l y i n c o r p o r a t i n g i t . be assumed that d e f a u l t r i s k i s - 156 - negligible. 2 THE ECONOMICS OF HUI WITH TWO OR THREE MEMBERS The purpose of this section is to p r o v i d e some b a s i c Hui by performing some p r e l i m i n a r y economic Hui examples. default explicit Hui Specifically, r i s k f r e e and of opportunity duration. n (=1, s i z e of 2, rate analyses on s e v e r a l that all type, which and that remain the here are a l l agents have an constant throughout with o n l y two or the three an o r g a n i z e r . section, we use the following the p e r - p e r i o d , b e f o r e - d i s c o u n t 3): simplified Hui c o n s i d e r e d Moreover, the Hui are s m a l l i n s i z e , F o l l o w i n g the p r e c e d i n g : the assume the d i s c o u n t - b i d interest members, with or without A we u n d e r s t a n d i n g of participant who succeeds deposit notations: into i n bidding for the the pool; pool at the nth p e r i o d ; 0 : the organizer who equal i n s t a l l m e n t s b^ : n receives of A at an interest-free each of the b i d submitted by p a r t i c i p a n t i b = max{b_£ }: i r^ : member i ' s n loan the N subsequent of NA r e p a i d i n N periods; at p e r i o d n; the h i g h e s t b i d submitted at p e r i o d n; opportunity interest rate, i = 1, 2, 3. Two-Member H u i Without An O r g a n i z e r Consider first cash flows f o r i t s a Hui with members 2 members are - 157 - and no o r g a n i z e r . The ex post time 1 2 member Suppose 1 A-b 2 "(A-b ) -A x A 1 member i ' s only alternative is to save w i t h banks at a r a t e r ^ . How much should he b i d at time 1? If he i s account. to r e c e i v e The ( g r o s s ) amount he has to money back at time bank return after pay at 2, time the proceeds w i l l go to h i s bank one p e r i o d must not be l e s s than A, the 2. If he i s to lend at the Hui l e n d i n g r a t e must be at saving rate r ^ . tions. funds at time 1, L e t v^ denote time 1 and get the equal to the least a b i d that s a t i s f i e s the above c o n d i - Then, (A-v )(l+r ) 1 ± = A, which i m p l i e s Ar. - A - J±— = i v i The v^ i n at time 1. loan A for earned (2.1) can be i n t e r p r e t e d as the Expression (2.1) one on A . period Let's a v a i l a b l e at time 1. dv. dr In f a c t , ± (2.1) i is says that the p r i c e ( i n time l ' s the present value c a l l v^ member i ' s Note ' p r i c e ' for getting of the a l o a n of A d o l l a r ) of a one-period interest r e s e r v a t i o n d i s c o u n t f o r the loan that . (I+rTT^ regardless of > °the size of membership, a Hui member's r e s e r v a t i o n - 158 - discount his for funds available opportunity interest at each rate period is r^ . an i n c r e a s i n g f u n c t i o n of We t h e r e f o r e have the following proposition: Proposition 2.1: A Hui member's r e s e r v a t i o n d i s c o u n t portunity interest Obviously, if and each knows be b e t t e r of off default please. the then there dual. Then, it is which i s op- form a H u i . will possibility any time they to an N-member H u i : i n d i v i d u a l s have an i d e n t i c a l o p p o r t u - a common knowledge to everyone's It rate shared by every indivi- advantage not to form a H u i . are known to each member? In this both members know t h a t is > easy to see = v that to member b i d under l's are = 10%. bid. incentive to lower his To i l l u s t r a t e Accordingly, this, v^ = $16.67 and rates of Hui On the other hand, member 2 have no i n c e n - I f member 2 wins w i t h a b i d between v^ and interest cost and would b i d over v^ o n l y i n an attempt worse-off. the ex post 2 member 1 has borrowing by b i d d i n g under v^. bers to of making withdrawals and both r^ and v tive no p o i n t be extended Suppose a group of rate is save w i t h the bank to a v o i d flexibility T h i s argument can e a s i l y What i f r^ > It fact, and r e t a i n the interest case, two members have the same o p p o r t u n i t y i n t e r e s t f o r both members to nity in his rate. this P r o p o s i t i o n 2.2: increases and p r o f i t s = $9.09. 159 , both mem- r^ = 20% and We can then calculate from Hui f o r both members depending on the a c t u a l winning b i d . - assume A = $100, to push up - T a b l e 2 . 1 : Ex Post I n t e r e s t Rates and P r o f i t s f o r A 2-Member Hui — An Example ex post member bids b x b 1 b l b l b 1 b 1 = 16.67 > b = 12.00 > b = 9.09 > b < b 2 < b < b 2 2 2 2 = 9.09 = 12 = 16.67 The p r o f i t the 2 lender, is cash flows interest rate profit* +83.33 -100 20% (borrow) 0 -83.33 +100 20% ( l e n d ) 8.34 +88 -100 13.6% (borrow) 5.6 -88 +100 13.6% ( l e n d ) 3.2 +90.91 -100 10% (borrow) 9.09 -90.91 +100 10% ( l e n d ) 0 -90.91 +100 10% ( l e n d ) +90.91 -100 10% (borrow) 0 -88 +100 13.6% ( l e n d ) -5.6 +88 -100 13.6% (borrow) -3.2 -83.33 +100 20% ( l e n d ) +83.33 -100 20% (borrow) -9.09 0 -8.34 ( A - b ) ( l + r ^ ) - A f o r the borrower and A - ( A - b ) ( 1 + r ^ ) f o r where b = max{b^,b }. 2 - 160 - It f o l l o w s from the a n a l y s i s P r o p o s i t i o n 2.3: i n Table 2.1 F o r a 2-member Hui with r^ > r^, f o r member 2 to win the only b i d d i n g at The bers' actual relative If that: split of the bargaining potential time it is examined the bidding profit depends on these two mem- positions. strategy i n Sections Pareto-optimal 1. each member knows o n l y the d i s t r i b u t i o n of h i s discount, not becomes much more rival's complicated reservation and w i l l be 3 - 5 . Two-Member Hui With An Organizer How would the an o r g a n i z e r , the introduction period 0 2A -A -A 1 -A 2A-b 2 -A "(A-b^ of + i> which i m p l i e s v rate r ^ , member i With following: -A x (i 2A = 1, r e t u r n equal to r ^ , which means h i s 2) w i l l demand at reservation discount not p o s s i b l e f o r a 2-member Hui to support an o r g a - that o^T ^ = P r o p o s i t i o n 2.4: interest picture? become the 2 a rate (who participants 1 v . would be such l nizer change the 0 Given the o p p o r t u n i t y A ( 1 + r an o r g a n i z e r cash flows f o r the participant least of _ It obtains " 2 A " v i , A r ^ / ( l + r ^ ) < 0. is Hence, an i n t e r e s t - f r e e loan) of r e t u r n to both members. - 161 - and yet yield a positive rate Three-Member H u i Without An O r g a n i z e r In t h i s case, the cash flows time member 2A-b 3 "(A-b^ "(A-b ) time 1 ( i . e . If his will v^) (A-v. )(l+r ) 2 (A-v.^d+r^ 2 V V i 2 =I i l Note =T (in A 2 t v^ 2 2 time the funds a v a i l a b l e v^)« r ^ , what w i l l be h i s r e s e r v a t i o n d i s - fall back p o s i t i o n ( i . e . the f o l l o w i n g o b t a i n i n g 2A at conditions: (2.2) (2.3) + (A-v (2.4) 2 1 2 )(l+r ) together = 2A. ± guarantee a r a t e of r e t u r n r^, and imply (2.5) 2 2 - l+T- - T l + r T T ^ i I does flows not depend on v ^ 2 pre-paid of and can be o b t a i n e d by s o l v i n g becomes c l e a r e r i f we take 2 ( - A + v ^ , 2 A - v ^ , - A ) over 2 In a sense, 6 <'> The i n t e r p r e t a t i o n of v ^ cash of the other at + A = (2A-v. )(l+r.), (0,3A-2v^ ,-3A). form for 1, = A d + r ^ + A, (2.4) and ( 2 . 4 ) . the time ^ -TTFH. A that incremental i.e., i - his should s a t i s f y 2 i (2.2) rate is Given 2(A-v. )(l+r ) 1 one at and another f o r funds a v a i l a b l e at time 2 ( i . e . 2 1 2A have a r e s e r v a t i o n d i s c o u n t and v ^ ? and v ^ -A 2 2 be two b i d d i n g s , opportunity interest Relations -A "(A-b^ there w i l l v^ 3 2 at time 3 ) , 2 -A x each member i (2.3) 1 2(A-b ) 2, v^ follows: 1 Now, because counts are as 2v^ interest) of - 162 2 (-A+v^^ , - A + v ^ , 2 A ) , 2 can be i n t e r p r e t e d as the p r i c e getting - the an i m p l i c i t l o a n of 3A at time 2, payable at time 3. 2v. To v e r i f y t h i s , rewrite (2.5) as (2.7): = 3Ar./(l+r ). 2 (2.7) i The RHS of (2.7) g i v e s the present v a l u e of the o n e - p e r i o d i n t e r e s t Similarly, (2.6) can be r e w r i t t e n as 3v i which o f f e r s 0. 2 (2.8) 1 a s i m i l a r i n t e r p r e t a t i o n for that, Moreover, benefit as v^ from the stated > v^ 2 to all members, then l o s s of g e n e r a l i t y Proposition 2.5: that, that at The rates that, there is if no r^ = r reason 2 = r^ i s for that r^ > r > r^. 2 r^ > r 2 > loan ... the P r o p o s i t i o n 2.3 > r^. Then, bidder with it the is can be flows ge- opportugene- advantage highest opportunity i n P r o p o s i t i o n 2.5 needs clarification. interest for which twice unique i n t e r n a l r a t e s of either rate' interest change - 163 - 2.1 a pure lender or a pure borrow- signs, return. For a g i v e n i n Table a Hui w i t h 3 or more members due cash and assume to the g r o u p ' s er. so a Hui. N , and assume without l o s s of a member i s haved' form rates, i s w e l l d e f i n e d because not i ^ first. term 'advantage' is d r below: 2-member H u i , both ' p r o f i t ' and 'ex post It 2 ' a common know- them to opportunity interest r^, i = 1 each p e r i o d , rate obtains y i Suppose the N members of a Hui have hetergeneous interest rality d v discount. n e r a l i z e d to P r o p o s i t i o n 2.5 nity ^ v ^ / d r ^ > 0 and due to the d e c r e a s i n g number of members who would Suppose members have heterogeneous without v^. i n Propostion 2.1, We know from P r o p o s i t i o n 2.2 ledge (2.8): = [3Ar +v. ]/(l+r ), n No t e on 3A. consequently to the might 'ill-behave non- For a Hui i n t e r e s t rowing r a t e s the to be d i s t i n g u i s h e d pure l e n d e r borrowing rate rate flow stream. cash flow funds at Hui l e n d i n g For instance, the rate, given his a n d Y^ r e s p e c t i v e l y , (A-^Xl+Yj) - last important that bor- For each member (except period), there are one Hui both d e r i v e d from the same cash rate and Hui borrowing and l e n d i n g rates, are such (2A-b ) - is opportunity interest (-A+b^ ,Zk-b^ , - A ) , member 2's denoted by it from l e n d i n g r a t e s . who o b t a i n s and one be m e a n i n g f u l , that ^ 2 and (2A-b ) - (A- 2 In other words, to borrows o n l y a f t e r b l )(l r ) = + calculate time he o b t a i n s As an example, funds; is Sections an rate r . 2 thereafter, r v i 1 20% he borrows at h i s 2 assume 2 3 be as v il 16 19.7 20.7 10 13.2 13.6 optimal bidding so t h a t their bids strategy he under under lends i n Hui up to opportunity rate. The follows: their a set reservation of discount assumptions. the a c t u a l winning b i d s are b^ = b - 164 - he i2 $25.0 shade that, that = 16%, r^ = 10% and A = $100. $23.6 bidders 3-5.) we S y m m e t r i c a l l y , he lends f o r each member w i l l i (This Hui borrowing r a t e , assume r^ = 20%, r reservation discounts Suppose his he has drawn funds from the p o o l ; b e f o r e at h i s o p p o r t u n i t y i n t e r e s t the . 2 2 = $20. See Then, the actual cash flows member are g i v e n and the Hui borrowing and lending rates for each below: time 20% This than h i s $160 himself + 180 - 100 10 - 80 - + 200 tells $180 a l o a n at ing one-period a special story. 20%. i n his Member 1 borrowed at Member 3 lends at Member 2's $80 for outcome period is loan with account, of than his If he more l i k e l y to The i n t e r e s t r a t e he he position. is 16.3%, h i g h e r value lower $100, than 16%. himself his If he as borrow- Hui borrowing than 16%. consider case of a maturity 16.3%, 15.8%, h i g h e r depends on h i s bank savings one 16.3% 15.8 time 2 f o r some purpose and c o n s i d e r e d 13.6%, l e s s We s h a l l not is 80 as having saved with Hui f o r one p e r i o d . needed is rate proceeds from l e n d i n g rate a happy 10%. earned a 13.6 80 opportunity rate see 16.3% - opportunity the -$100 15 table put -$100 3-member Hui w i t h an o r g a n i z e r the model to be s t u d i e d - 165 - i n Sections here 3 - 5 . because it 3 THE MODEL FOR AN N-MEMBER HUI WITH AN ORGANIZER Notations As A i n Sections 1 and 2, we w i l l use the f o l l o w i n g : the s i z e of the p e r - p e r i o d , b e f o r e - d i s c o u n t n (=1, notations: deposit into the p o o l ; N ) : the p a r t i c i p a n t who succeeds i n b i d d i n g f o r the p o o l at the nth p e r i o d ; 0 : the organizer who receives equal i n s t a l l m e n t s b^^: of A at each of the b i d submitted by p a r t i c i p a n t i b^ n : the h i g h e s t b : the second h i g h e s t n an I n t e r e s t - f r e e loan of the N subsequent NA r e p a i d i n N periods; at p e r i o d n; b i d submitted at p e r i o d n; Other n o t a t i o n s b i d submitted at p e r i o d n . will be d e f i n e d and e x p l a i n e d as they arise. Assumptions Unless otherwise stated, the following assumptions are made t h r o u g h - out the paper. Assumption 0: Assumption 1: There i s There no p o s s i b i l i t y of is no Bayesian default. learning from past winning bids by i n d i v i d u a l members i n d e c i d i n g t h e i r c u r r e n t b i d s . Assumption 2: Each individual i has a deterministic and known income stream 1^ over the d u r a t i o n 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 O ' ! ! i N ^ ' » » •**» " I = 1 I 1 = 1 - 2 166 N - Assumption 3: von Each individual i has a continuous Neumann-Morgentern u t i l i t y function and s t r i c t l y defined increasing over his income stream. Assumption 4: To each b i d d e r i , n are drawn i n d e p e n d e n t l y Tb ,b L tegy, and investigate 'positive time Definition 3.1: from a l l o t h e r b i d d e r s at p e r i o d w i t h support 1. n J utility both the maximizer's optimal bidding s t r a - 'reservation price' i n the context of Hui preference': The r e s e r v a t i o n d i s c o u n t v e c t o r v^ of p a r t i c i p a n t i v v vector v^ = ( v ^ » ^ 2 ' ' ' ' > ± to N alternative the -n' an expected we need to d e f i n e bids from p r o b a b i l i t y d i s t r i b u t i o n c o n t a i n e d i n some i n t e r v a l To the words, Uj^Y.^) = U cash N-l^ flow s u c n t n a t patterns f o r a l l n = 1, 2, participant i listed is is any indifferent i n Table 3 . 1 . N, In other (3.1) where Y Y Y We il = in = iN = (I 0 I + - » il N A N 1 v I (I o- > ii-( - ii)."-» A A I A I A I ( iO- » il" shall refer I -( - ) il» i2" '-"' iN" v I 1 b e g i n n i n g of ( to A - v ) + N A l n -C - ) I N ll »"-' l,N-l- as ( n A v A ) ' I l n v A »---» i - )» N ) I - i,N-l » iN' participant i's W A ) - r e s e r v a t i o n u t i l i t y at the the H u i . D e f i n i t i o n 3.2: his A i An agent i preference for is said to income streams - exhibit of 167 positive the f orm - time p r e f e r e n c e ( X Q , . . . ,x^) is such if that T a b l e 3.1: P a r t i c i p a n t i ' s I n d i f f e r e n t i n a D i s c o u n t - b i d Hui period •. 0 1 0 NA -A 1 -A A + ( N - l ) ( A - v Cash Flow P a t t e r n s n • N-l N alternative • n -A i;L ) .•• — A. • -A • -A -A -A -A • "(A-v ) i l • .. n A + ( N - n ) ( A - v ) . -A ln -A • N-l -A -(A-v ) -(A-v ) .. (N-DA+CA-v^j,^)-A N -A -(A-v ) -(A-v ) -< - i,N-l> 1 ] L ± 1 i n i n - 168 - A v NA ( X Q , . . . , X J ...,x^) if A set is x^,...,x^) i is s t r i c t l y p r e f e r r e d to ( x ^ , . . . , x ^ , . . . , x ^ , and o n l y i f x^ > x^ f o r h = 0, of s u f f i c i e n t conditions 1, N and k > h . f o r the e x i s t e n c e and uniqueness g i v e n i n Lemma 3 . 1 . Lemma 3 . 1 : The r e s e r v a t i o n d i s c o u n t v e c t o r continuous, bits time proof discount the exhi- preference. of vector, Lemma 3.1 which a l l entries is involves then except the nth and the nient backward c o n s t r u c t i o n of N and N - l pends only the shown to i n every two adjacent tical ods unique i f See the A p p e n d i x . The 3.1, e x i s t s and i s s t r i c t l y i n c r e a s i n g von Neumann-Morgenstern u t i l i t y positive Proof: to ,, , be u n i q u e . rows, ( n + l ) t h ones. • which i n words, as v ^ f ° auctions r are conducted This feature p e r i o d k. that, depends allows in Table are i d e n the conve- s t a r t i n g with constructed, on v . ,„, .... and b i d s revealed at periods a l l k > n does not change as l o n g as b i d d e r i ' s p r e v i o u s l y mentioned termediate reservation peri- v^ de- In other i,n+z income stream remain unchanged. utility a say rows n and n+1, From the way v^ i s turn of Note reservation discounts, o b t a i n v^ on v . construction i,n+1 n, of Consequently, also applies We may c a l l it to 1, 2 p r e f e r e n c e and the p r o p e r t y of reservation the Hui cash flow at any I n - member i ' s conditional reservation -k utility at ...,c, ,). k-1 U i ( Y p e r i o d k, denoted by , g i v e n some r e a l i z e d cash flow ( C Q , C ^ , In other words, i,k+t ) = U ±» fc = °> 1 » •••» N k ~ « where - 169 - ( 3 ' 2 ) Y ' i,k+t = ( c c o' i'--- j C k-i > I ik" ( A v ) ~ ik '*", 0 ( 3 ^.k^^-^-^.k^^i.k+t+r^-- The •••» J 3 ) c o n d i t i o n a l r e s e r v a t i o n d i s c o u n t v e c t o r v, then r e f e r s to (v,, , i ik » 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, V I, N-1 The vidual an - 0 r e s e r v a t i o n d i s c o u n t has a f a m i l i a r i s r i s k n e u t r a l with interest ticular rate available case, individual time 1 k-1 i n t e r p r e t a t i o n when the i n d i - p r e f e r e n c e being completely i n some formal financial i ' s reservation discounts sector. d e s c r i b e d by In t h i s represent par- h i s market- based o p p o r t u n i t y c o s t of Hui d e a l i n g , and can be o b t a i n e d by equating the present 3.1. values of the N a l t e r n a t i v e cash flow listed i n Table By t e n d e r i n g b i d s equal to h i s r e s e r v a t i o n d i s c o u n t s a t each p e r i o d , a Hui p a r t i c i p a n t vation u t i l i t y , ket. In t h i s back' position can at l e a s t attain a u t i l i t y utility sense, with respect to j o i n i n g the H u i . It i s clear 'fall that, i f a the d u r a t i o n , h i s ex post a Hui w i l l never be l e s s than h i s ex ante r e - has no access to the f o r m a l f i n a n c i a l market, or utility. When an i n d i v i d u a l the formal financial market can be i n t e r p r e t e d as r e l a t e d reflects equal to h i s r e s e r - r e s e r v a t i o n d i s c o u n t s d e s c r i b e an i n d i v i d u a l ' s from p a r t i c i p a t i n g servation level which i s what he can o b t a i n from the f o r m a l f i n a n c i a l mar- member b i d s h i s r e s e r v a t i o n d i s c o u n t s throughout if patterns his 'internal' does not e x i s t , to some s o r t opportunity cost consumptions. - 170 - h i s r e s e r v a t i o n discounts of 'as i f i n t e r e s t r a t e which such as time preference f o r 4 OPTIMAL INDIVIDUAL It level, BIDDING STRATEGIES is shown i n S e c t i o n called 'reservation 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 utility', servation discount. ximizer' s optimal bidding s t r a t e g i e s . impose, and v i a Assumption 3 ' , positive is we c o n s i d e r an expected To make the on i n d i v i d u a l s ' strictly positive time increasing, the following preferences. concave, of each individual i , and t i m e - a d d i t i v e easy to see equivalent to decreasing, 1 With that strictly (4 requiring the positive time time p r e f e r e n c e preference is, coefficients ' in this to be 1} case, strictly i.e., X 1 0 > X u > ... > X time-additive 1 N > 0. utility, ward through dynamic programming. last p e r i o d N, the There i s function way: = is with preference. VV-'V Co^inW' It we time-additivity, We can w r i t e a t i m e - a d d i t i v e von Neumann-Morgenstern u t i l i t y in re- u t i l i t y ma- problem t r a c t a b l e , The von Neumann-Morgenstern u t i l i t y continuous, (strictly) section, the requirements of c o n c a v i t y , time p r e f e r e n c e Assumption 3 ' : U^, In t h i s by s u b m i t t i n g a b i d equal to h i s (4.2) our problem can be From Table 1.1a, only member who has not no need f o r b i d d i n g and t h e r e f o r e proceed backward to p e r i o d s N - l , N-2, - yet solved we note received working backthat, - the funds o b t a i n s NA. no u n c e r t a i n t y i n v o l v e d . . . . , the number of b i d d e r s 171 at As we increases by one each time. At each p e r i o d , the prospect of l o s i n g e n t a i l s pected u t i l i t y g a i n from p a r t i c i p a t i o n i n subsequent periods. the ex- In g e n e r a l , t h e r e a r e N-n+1 b i d d e r s at p e r i o d n , each of whom submits a b i d t h a t m a x i mizes the h i s current potential expected subsequent utility expected which i n c o r p o r a t e s , utility g a i n from i n a nested way, future periods. The a c t u a l form of the expected u t i l i t y f u n c t i o n depends on the a u c t i o n i n g method ( i . e . f i r s t - p r i c e or s e c o n d - p r i c e ) . of f i r s t - p r i c e competitive 4.1 We s h a l l f i r s t c o n s i d e r the case b i d d i n g as the a l l o c a t i o n mechanism. F i r s t - P r i c e Competitive Bidding In a f i r s t - p r i c e sealed b i d Hui where Assumptions 0 (no d e f a u l t ) , (no l e a r n i n g ) , 2 ( d e t e r m i n i s t i c and known p r e - H u i income s t r e a m ) , tinuous, strictly Morgenstern pendent increasing, utility with and i d e n t i c a l expected E U concave, strictly and t i m e - a d d i t i v e positive time agent) and 4 ( i n d e - hold, u t i l i t y at p e r i o d n of a b i d b ^ i s g i v e n by e x p r e s s i o n n b in< in> 3' ( c o n - von Neumann- preference) b i d d i s t r i b u t i o n f o r each 1 bidder i ' s (4.3): WnV^n^-^-^in) • + C in ^ i n - i ^ i n - ^ ^ ^ i ^ l ^ ^ i n ^ ) ^ " 1 1 ' 4 < * 3 ) where E U * f i,n+l The f i r s t = E u f ..(b* i,n+l ..) i,n+l = Max Euf i,n+l term of the RHS of e x p r e s s i o n der i i f he wins a t p e r i o d n , whereas - (b, v ..). i,n+l (4.4) ( 4 . 3 ) i s the u t i l i t y f o r b i d - the second term g i v e s h i s u t i l i t y i n 172 - the case of his utility losing. D The component from p o s t - b i d d i n g i X ^ u ^ I ^ - A + x ) , where x e ( ^ > ^ ] » n income at the c u r r e n t p e r i o d . s n EU*^ , , is i,n+l independent utility at of the the winning next period bid at providing u t i l i t y maximizing b i d at each of Sufficient ted utility define the conditions globally terms are utility function is D e f i n i t i o n 4.1: F is for that n and r e p r e s e n t s the expected bidder his expected subsequent an i n d i v i d u a l ' s stated outbidden in rate' distribution', i submits periods. b i d to maximize Theorem 4.1 and which expec- below a f t e r 'increasing ensures the we (decreasing) that the expected pseudo-concave. The m a r g i n a l o u t b i d d e n r a t e 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 g i v e n by F ' / F . An interpretation shortly after Definition of the A probability marginal In Theorem 4 . 1 , of dual's rate will be Suppose decreasing expected rate is the f e a r of Assumptions called yields an an increasing increasing (decrea- i n d i v i d u a l and time s u b s c r i p t s 0, 1, maximizing 2, 3' bid g i v e n by the s o l u t i o n at and 4 h o l d . Then, distributions, the nth the period, and i s F'(b) F(b) u'(I+NA-(N-n)b) [u(I+NA-(N-n)b)-u(I+NA-(N-n)v)]-[u(I-A+b)-u(I-A+v)+E*] to the f o l l o w i n g with - ( i and ambiguity. marginal-outbidden-rate utility that uniquely; = provided distribution. we suppress to F , b, I and v without class outbidden distribution outbidden sing) marginal-outbidden-rate Theorem 4 . 1 : marginal Theorem 4 . 1 . 4.2: (decreasing) n) the formally 'marginal marginal-outbidden-rate period 173 - b, for the indiviexists equation: „ K J E* = d / ^ ^ C E U ^ ^ - u f ) , 1 is where EU* i,n+l tional Proof: given i n expression is difference between the utility at period n, ness of the expected first is decreasing The t i o n as ginally' not A first expected which, order condition utility and the by c o n s t r u c t i o n , is u t i l i t y maximizing b i d i s satisfied by assumption, and t h a t for bidder i ' s condi- outbid by A decreasing his bid, be slower common established of second rival increase in increase decreasing the The u n i q u e - by showing and the RHS i n c r e a s e s p r o b a b i l i t y that rate chance i n the that he means of chance i n b by the satisfied has the locally. interpreta- his that as an i n d i v i d u a l first rival must being m a r g i n a l l y o u t b i d . marginal-outbidden-rate a special 'mar- outbid distributions 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 b u t i o n used i n S e c t i o n 5 i s con- has outbidding his of that equation an i n d i v i d u a l b i d d i n g b i s given the reservation u n i q u e l y ( s i n c e the LHS of m a r g i n a l outbidden than the class g i v e n by the the his maximize a constant. the second o r d e r c o n d i t i o n i s conditional b to conditional d e c r e a s i n g m a r g i n a l outbidden r a t e F ' ( b ) / F ( b ) the rival. the order c o n d i t i o n i s c a v i t y of u ) , raises is See the Appendix. (4.5) (4.5) -n+l and U (4.4) r e s e r v a t i o n u t i l i t y at p e r i o d n + l . Equation the (4.6) distri- case. When b i d s are u n i f o r m l y d i s t r i b u t e d and b i d d e r s are r i s k n e u t r a l , individual solution expected (given discussions in on t h i s utility maximizing b i d w i l l have expression special (5.3)). We case i n S e c t i o n - is 174 - 5. will a closed-form have more the solu- detailed 4.2 Second-Price Competitive The formulation second-price with minor (i.e. sealed Bidding of a c o m p e t i t i v e b i d d i n g model f o r the h y p o t h e t i c a l b i d Hui i s s i m i l a r modifications to that to i n c o r p o r a t e implemented) d i s c o u n t of the f i r s t - p r i c e the f e a t u r e i s the second h i g h e s t that bid. model, the e f f e c t i v e Under Assumptions 0, 1, 2, 3' and 4, the problem of the expected u t i l i t y maximizer i a t per i o d n i s s t a t e d below: M a X E U in ( b in> = -n (N-n){\ u (I + n i (N-n)C + J* { X in i n i n l n -A b + u (I 1 i n l n ) EU| + S n + 1 }[F -A y) EU^ + + n + 1 i n N (b }d[F i n n )] - - N i n 1 [ n l-F i n 1 (y)] - - dF i n (b i n ) ] (x), (4.7) where S EU* = EU® ^ ( b * i,n+l i,n+l i,n+l The first term i n e x p r e s s i o n succeeds i n b i d d i n g . out = Max EU® (b. . . ) . i,n+l i,n+l (4.7) i s the u t i l i t y The second term i s h i s u t i l i t y to be the second h i g h e s t bidder. The l a s t (4.8) f o r bidder i f he f a i l s i i f he and turns term accommodates the case where h i s b i d i s the t h i r d or below. Suppose an i n d i v i d u a l who b i d s b knows that he has been o u t b i d by one of h i s r i v a l s . I t i s then to h i s advantage to be o u t b i d by another so that he, as w e l l as other Parallel a to the 'marginal 'marginal Definition outbidding l o s e r s , can b e n e f i t from a h i g h e r rival, second b i d . outbidden r a t e ' i n the f i r s t - p r i c e case, we need r a t e ' f o r our s e c o n d - p r i c e 4.3: The m a r g i n a l outbidding rate - 175 - model. for a probability distribu- tion F is g i v e n by 1 F (t)/[l-F(t)] 'failure that rate' is (or In has the the fail bidding at time context, interpretation bidding outbid 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 'hazard r a t e ' ) , a product w i l l t. tion F'/(l-F). of the the is first rival. single-peaked, creasing marginal probability in this + is given that marginal his outbidding the rate. d i s t r i b u t i o n with it has s u r v i v e d up to rate rival that the is greater easy an i n d i v i d u a l that he has expected u t i l i t y to check that than or equal time F'(b)/[l-F(b)] that given the probability b i d d i s t r i b u t i o n F to y i e l d It index conditional probability second To make sure we r e q u i r e the outbidding conditional b w i l l marginally outbid by h i s t which as to funcan i n - the one been power belongs category. Definition 4.4: creasing) 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 marginal outbidding rate is called yields an i n c r e a s i n g an i n c r e a s i n g (de- (decreasing) marginal-outbidding-rate d i s t r i b u t i o n . After set of pected suppressing sufficient utility Theorem 4 . 2 : class of dual's F'(b) _ l-F(b) subscripts conditions for i and n, we s t a t e an i n d i v i d u a l ' s i n Theorem 4.2 b i d to maximize h i s a ex- globally. Suppose increasing expected uniquely; the and i s Assumptions 0, 1, 2, 3' marginal-outbidding-rate utility maximizing g i v e n by the bid solution at and 4 h o l d . Then, distributions, the nth period, to the f o l l o w i n g for the the indivi- b, exists equation: -u'(I-A+b) [u(I+NA-(N-n)b)-u(I+NA-(N-n)v)]-[u(I-A+b)-u(I-A+v)+E*] , (4.9) with E* = d / ^ X K U j ^ - u f 1 ) , (4.10) - 176 - E where U S * n + ^ i- s given i n expression (4.8) and is his conditional r e s e r v a t i o n u t i l i t y at p e r i o d n + l . Proof: Omitted. The proof of Theorem 4.2 can first (4.7), obtain first then show t h a t , bution, the first also satisfies 4.3 the would be s i m i l a r to t h a t of Theorem 4 . 1 . order condition for a problem e q u i v a l e n t with an i n c r e a s i n g m a r g i n a l - o u t b i d d i n g - r a t e order condition is uniquely satisfied We to distri- by a b i d which the second order c o n d i t i o n . Implications In this 4.1 and 4 . 2 , the following literature subsection, making r e l e v a n t that 4.1: bidders w i l l Under the Suppose equation (4.5) this is Implications shade the of Theorems First of all, standard t h i n k i n g i n the b i d d i n g their bids in f i r s t - p r i c e auctions re- behavior i n a H u i . hypotheses u t i l i t y maximizing b i d i s Proof: several comparisons when a p p r o p r i a t e . c o r o l l a r y t e l l s us that mains true of a g e n t s ' Corollary we w i l l d i s c u s s is less not Theorem 4 . 1 , than h i s true. negative. of reservation Then u | But the an a g e n t ' s discount. > 0 implies LHS of the expected that equation is the RHS of positive. This is a c o n t r a d i c t i o n . Q.E.D. The ing, well-known however, result does not that second-price in general auctions are demand reveal- apply to Hui a c c o r d i n g to the c o r o l l a r y below. - 177 - Corollary 4.2: Under the maximizer w i l l expected shade h i s utility than compensate hypotheses gain his of Theorem 4 . 2 , b i d under h i s from the utility next loss at reservation and the an expected discount subsequent utility only i f periods the will more c u r r e n t p e r i o d from shading his bid. Proof: Omitted. In a f i r s t - p r i c e bid i n excess of his sealed reservation he o b t a i n s is not affect what he has the winning price sealed count. his and to pay. the bid, the b i d might less he w i l l to the will a point outweigh own b i d . have to be the pected beyond which h i s the utility If he l o s e s , pay. b i d over h i s Because He, however, If from subsequent utility laries 4.3 and Corollary 4.3: pected lower highest the sethat to his bid only biddings high ex- even b i d under his the l a s t f i n a l period. second Hence, periCorol- 4.4. Under the hypotheses of u t i l i t y maximizer i n a f i r s t his dis- a chance sufficiently he might of the is g a i n from f u t u r e to at on the will inflate od due uncertainty second- he has some i n c e n t i v e he p e r c e i v e s periods, a the h i g h e r there O b v i o u s l y , he w i l l o v e r b i d at absence h i s b i d does reservation depends reservation discount. the the amount an i n d i v i d u a l i n highest, expected to due to the d i s p a r i t y between When he l o s e s , second current benefit. gain to no i n c e n t i v e s when he wins, amount he o b t a i n s b i d over h i s r e s e r v a t i o n d i s c o u n t . up to bid. discount, some i n c e n t i v e than h i s t u r n out since, In c o n t r a s t , effective b i d Hui has bid rather discount f u n c t i o n of h i s When a b i d d e r wins, rejected cond a decreasing bid b i d H u i , an i n d i v i d u a l has bid at the current period - 178 Theorem 4.1 (Theorem 4 . 2 ) , (second-) p r i c e sealed if - his perceived an ex- b i d Hui w i l l expected utility g a i n i n the f u t u r e Corollary 4.4: Under maximizer w i l l last the tend to Theorem 4 . 2 , than h i s the o p p o s i t e the a u c t i o n i n g methods, d r i v e down the rent bidding w i l l an expected but t h i s effect is value present bid. i n the unambiguously lower each ly their in the reservation on loans being reservation discounts, sense that by the the highest provement. allocation the l o a n always reservation On the c o n t r a r y , i f upon Pareto-efficient It is which no than h i s bid results second illustrate g a i n i n the cur- correctly auctioned off a l l o c a t i o n of goes to discount, measure each p e r i o d . loans i.e., will bid be improvement to if exact- efficient the most needing agent leaving the Then, (indica- no room f o r Pareto i m - someone e l s e wins the l o a n , the most 'purchase' can be the made need- loan. is called An a a l l o c a t i o n of the l o a n p r o v i d e d w i t h i n the H u i . auction is competitive not i n Hui a u c t i o n s ? period in bidding l i t e r a t u r e a Pareto-efficient w h i l e the demand-revealing s e c o n d - p r i c e parallel bid- reservation discounts discount), Pareto w e l l known i n the sealed from f u t u r e T h e r e f o r e , a member's b i d i n i n g person can presumably make a side-payment last the f i r s t - p r i c e H u i , but has b i d d e r s f o l l o w the demand-revealing b i d d i n g s t r a t e g y , price at the case f o r the s e c o n d - p r i c e H u i . bidder's he p l a c e s gains The p o s s i b l e i n the s e c o n d - p r i c e H u i . not n e c e s s a r i l y Suppose ted utility reservation discount possible push the b i d even lower f i r s t - p r i c e Hui i s all of second p e r i o d . dings true hypotheses submit a b i d h i g h e r R e g a r d l e s s of the increases. allocation sealed b i d auction i s . The answer a second-price that Hui). this. - 179 - is negative We c o n s t r u c t a firstmechanism Do we have (except f o r an example the to Example 4.1: Suppose (i) individuals bef o r e - d i s c o u n t loan averse individual where whereas X = l/(l+r), lending interest r b i d s and ex post a first-price is rate, come stream ( 1 0 0 , 0 ) , 2 is Hui, (ii) risk individual n e u t r a l with 2's non-Hui ( i i i ) both i n d i v i d u a l s have (iv) their respective incomes are g i v e n risk 2 2 borrowing as well as the same p r e - H u i i n - reservation discounts, actual below: 1 25 12 ( +16,+100) 2 20 16 (+184,-100) of r e s e r v a t i o n 1 is u ( x ^ , x ) = x^+Xx , reservation discount the d e f i n i t i o n f o r a $100 individual individual (winner) By in 1 and 2 are competing bid discounts ex post income and the dominance argument, we have (+175,-100) ~ (+25,+100) (+184,-100) >- (+16,+100) for individual 1, >- (+180,-100) ~ (+20,+100) for individual 2. Suppose i n d i v i d u a l the loan, 1 makes a side-payment the r e s u l t i n g income (+175,-100) and (+25,+100), streams of $9 to i n d i v i d u a l for individuals respectively. 2 to 'buy' 1 and 2 w i l l be 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 b e t t e r S i m i l a r examples can be c o n s t r u c t e d f o r a s e c o n d - p r i c e Hui w i t h N > 3. 1 Hence P r o p o s i t i o n 4.1 Proposition 4.1 4.1: This below: The expected and 4.2 w i l l credits off. utility not i n g e n e r a l maximizing b i d s yield given a Pareto-efficient i n Theorems a l l o c a t i o n of among H u i p a r t i c i p a n t s . non-Pareto-efficiency comes the f i r s t - nor the s e c o n d - p r i c e sealed from the observation b i d Hui e x h i b i t s - 180 - that neither the demand-reveal- ing property. bid depends ferences expect ever, The amount by which an i n d i v i d u a l w i l l on h i s and expectations the h i g h e s t by Corollary Corollary 4.5: Proof: When i n d i v i d u a l s ' allowed we can not below, to differ, the h i g h e s t we can pre- general reservation discount. expect reservation discount in an individual to his How- submit a increases. An i n d i v i d u a l ' s o p t i m a l f i r s t - (second-) (Theorem 4.2) increases Smith showed p r i c e sealed in his bid, current-period re- discount. See the A p p e n d i x . Cox, Roberson identical and preferences bid auction w i l l also 4.5 i n Theorem 4.1 servation are and e x p e c t a t i o n s . b i d d e r to have h i g h e r b i d when h i s given preferences shade or i n f l a t e and expectations, be P a r e t o - e f f i c i e n t . true f o r Hui a u c t i o n s . Corollary first- 4.6: (1982) Under mechanism (i.e. utility tions (i.e. for the the C o r o l l a r y 4.5 hypotheses credit functions of all bidders first-price t e l l s us that all time bid d i s t r i b u t i o n s at have sealed this is 4.6: Theorem 4.1 (Theorem 4 . 2 ) , b i d Hui w i l l be a P a r e t o - e f f i c i e n t if and when ordinary Hence, C o r o l l a r y (second-) p r i c e s e a l e d tion that, individuals preference possess the coefficients) the c u r r e n t and each of alloca- same and the the tastes expecta- subsequent periods). It on the a was mentioned p r e v i o u s l y t h a t c o m p o s i t i o n of member's its members. time preference affect higher preference for equal, a higher time p r e f e r e n c e a h i g h e r b i d f o r loans the a c t u a l outcome One q u e s t i o n his optimal current coefficient for 181 Other is: How does things (characterized the c u r r e n t p e r i o d ) - a Hui depends interest bid? consumption currently available. - of of The r e s u l t that being by should l e a d supports a to this intuition is formally C o r o l l a r y 4.7: stated i n C o r o l l a r y Under the hypotheses of Theorem 4.1 vidual optimal lative time p r e f e r e n c e Proof: See 4.7. bid i n a f i r s t - (second-) for current - p r i c e Hui i n c r e a s e s consumption the Appendix. 182 - (Theorem 4 . 2 ) , increases. an i n d i as h i s re- 5 A NASH PROCESS OF INTEREST RATE FORMATION Having with a derived the time-additive i n d i v i d u a l optimal implications i n S e c t i o n 4, the winning of u t i l i t y bers of their the rates which are derivation which w i l l addition yield to of the 1982; may be appropriate the tractability, interest the Nash the Smith and Walker, more plausible determine the we w i l l demonstrate utility by e x p r e s s i o n U.(x.) i i individual defined (5.1) f o r Hui mem- The definitions be 4 w i l l lead over h i s i risk income programs interaction the c o m p e t i t i v e In has bidding (Cox, Roberson and mechanism (sealed bid) bid- to a Nash e q u i l i b r i u m under The r i s k n e u t r a l i t y r e p l a c e s Assumption is given that the o p t i m a l i n d i v i d u a l and l i n e a r b i d d i s t r i b u t i o n s . Each how anonymous. risk neutrality Assumption 3": of F u r t h e r m o r e , the Nash assump- auctioning to remain s t a t e d below as Assumption 3", strategic object auctions derived i n Section v the q u e s t i o n Hui context w i l l s t u d i e s of ding strategies tion, some of i n an endogenous manner. of 1982). given used by H u i , which h e l p s members rate model as m u l t i p l e Cox, In t h i s s e c t i o n , in an agent Nash e q u i l i b r i u m i n d i v i d u a l b i d d i n g i n experimental b e h a v i o r f o r s i n g l e as w e l l tion maximizing agents w i l l relevant found e m p i r i c a l support Smith, we are ready to e x p l o r e for and s t u d i e d r e s p e c t i v e i n t e r e s t r a t e s ex ante and ex p o s t . interest after strategy von Neumann-Morgenstern u t i l i t y its bids bidding neutral with stream x^ = (X^Q, assump- 3'. the time-additive . . . ,x^) and given below: N = £ . \ , x, , n=0 i n i n ' (5.1) - 183 - 1 = X. where > X., > . . . n iO We First, have it ii two helps iN justifications us avoid tions of small compared with that actual interval either of the lation looking at (4.9)) firstbe tribution is neutrality function that price linear the in 4': Each individual at p e r i o d n , v ^ , serves his is Second, tend the to be observa- relatively pre-Hui incomes cluster tend to use where by e x p r e s s i o n discount. reservation independently n n = 1, v n w l» interval [b mone- the (4.5) Since individual (expression a uniform d i s - we can r e p l a c e Assump- n e r e discount for the 4' implies N-1. that the nth period, b drawn from a common uniform n = 1, N-1; F u r t h e r m o r e , each v. ,b. ], where b each i n , i.e., before individual i he submits individual 184 correspond, - ob- his b i d b. . in i's bid for the fund w i l l be u n i f o r m l y d i s t r i b u t e d over and b - fund (5.2) own r e s e r v a t i o n d i s c o u n t Assumption at This in their bid calcu- in available so within a a good a p p r o x i m a t i o n . transformation, i's is n i, or individuals' 4'. n all bids u t i l i t y worth, reservation d i s t r i b u t i o n H w i t h support [ Y » H (v. ) = (v. - v ) / ( v - v ) n in i n —n n —n for essay. Hui p a r t i c i p a n t s bid given preserved v i a a l i n e a r available this in and uniform b i d d i s t r i b u t i o n s , (second-) t i o n 4 by Assumption Assumption attitude assumption. 1981). risk will of available r a t h e r than some s u b j e c t i v e Under risk neutrality gambles whose outcomes f o r which a l i n e a r (Huang, optimal the risk discount loan seems to a c c o r d w i t h o b s e r v a t i o n s tary value, the the main focus dispersions we are v i r t u a l l y small for confounding b i d d i n g behavior which i s the > X . „ > 0. through a l i n e a r the rela- tionship, to and v , r e s p e c t i v e l y , different b i d d e r s might p e r c e i v e order produce to believes from that future Since every other biddings as an a g e n t ' s preference expected formally gain from tingent expectation, his future has own, gain we gains is shall from f u t u r e b i d d i n g s . we w i l l the same assume on discounted using bidding to it that discounted contingent refer because as losing his each this gain period. appropriate subjectively participation. agent expected at In This time discounted restriction is s t a t e d as Assumption 5: Each i n d i v i d u a l i discounted on l o s i n g We s h a l l expected believes gains n i n periods that all other bidders' Note t h a t n+l, G£ n subjectively is n+2, subjec- from f u t u r e b i d d i n g p a r t i c i p a t i o n , c o n - i n the c u r r e n t b i d d i n g , are equal to h i s denote member i ' s p e r i o d n by G* place different bidder expected coefficient, Assumption 5: tively consistent and are i n d i v i d u a l - s p e c i f i c his N-1, discounted expected gains subjectively and t h a t G | Q can be i n t e r p r e t e d as member i ' s expected gain from b i d d i n g s discounted expected own. at taking to p e r i o d n , 'surplus' for join- i n g the H u i . Given the m o d i f i e d assumptions, for the first-price 5.1 and Theorem 5 . 2 , and s e c o n d - p r i c e respectively. n+1 denote the number of e l i g i b l e funds) the Nash e q u i l i b r i u m b i d d i n g programs sealed b i d Hui are given To s i m p l i f y n o t a t i o n s , i n Theorem we l e t b i d d e r s (those who have not yet m = Nreceived at p e r i o d n . Theorem 5 . 1 : Suppose Assumptions 0, 1, 2, 3", 4' Nash e q u i l i b r i u m b i d d i n g program f o r i n d i v i d u a l i ed b i d Hui i s g i v e n by - 185 - and 5 h o l d . Then, in a f i r s t - p r i c e the seal- b *in = i n - 5TT < in-V " 5" l n > V where V G*^ is G the s u b j e c t i v e l y 5 3 < ' > d i s c o u n t e d expected g a i n from f u t u r e bid- ding p a r t i c i p a t i o n , g i v e n i t e r a t i v e l y by G l n = ^i.n+l^in^^^W^l^l' V b b b m ^ i,n r !,n l^ i,n r n l> " + + / for n = 1, Proof: * fi , \n+1 , ( X + 2 + V X b - i,n l^*i!n l^ - n l) " + m + 3 d x + 5 4 <'> l' . . . , N-l. See the Appendix. The for 2, + proof involves identifying the b i d f u n c t i o n ( 5 . 3 ) 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 Theorem 5 . 2 : Nash Suppose Assumptions 0, equilibrium sealed bid Hui i s h fi n bidding 1, program 2, for is 3", indeed the 4' as a c a n d i d a t e case. and 5 h o l d . individual i in a Then, the second-price g i v e n by: = .i n + -m+1 r r (v n-v, i n) - - mG *i n, ' v S (5.5) g where G* in is the subjectively discounted expected gain from future b i d d i n g p a r t i c i p a t i o n , g i v e n i t e r a t i v e l y by: S G * = (K in i,n+1 m in 1 {(m-2)/^ -n+1 + 1 (x-b i,n+l n = 1, 2, J 1 + for 2 )[(m-2)/(b , . - b , ) ~ ] * n+1 -n+1 m n + 1 3 ) - (v + 1 -x)dx ^ / ^ i . n + l ^ t n + l ^ ^ n + l ^ ^ ^ i,n+l N-l. - 186 - 5 < ' 6 ) Proof: Omitted s i n c e it is s i m i l a r to the proof of Theorem 5 . 1 . We summarize i n Table 5.1 well as the discounts est the Nash b i d d i n g s t r a t e g y for period n, values of c o r r e s p o n d i n g expected (i.e. winning b i d under the b i d under the s e c o n d - p r i c e and v a r i a n c e s the effective f i r s t - p r i c e method and second the in a single-object second-price variance results all is higher do not m > 2, auctioning ° all i, then under variance the is higher method. the of the expected methods It the we expected the is same, but discount (i.e. price the method w i l l second-price first-price discount of standpoint, discount the seems reasonable method, since from the to the the will same v a r i a n c e . a u c t i o n i n g method must be f o l l o w e d for b i d that is first-price be h i g h e r under the yields point of T h e r e f o r e , from a that view, a higher given members, se- the first- When m = 2, dominates expected the effective the m u l t i - p e r i o d nature and the f a c t that the throughout a c y c l e , same it is l i n e of argument to favor one method another. Since will the conjecture consistently not p r e s e n t l y c l e a r how to c a r r y t h i s check t h a t , s e c o n d - p r i c e method. However, H u i , the double r o l e p l a y e d by i t s over to lender's former similar f f s s that G* = G* = G* = G* f o r in n n in f u r t h e r assume be p r e f e r a b l e method with it corresponding than under the c o n d - p r i c e than under the f i r s t - p r i c e method f o r a l l m. borrower's the s t r a i g h t f o r w a r d to second-price effective showed p r i c e s under the f i r s t - and In our Hui b i d d i n g model, effective under the If are latter. appear o b v i o u s . the implemented) auction, auctioning high- 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 , V i c k r e y (1961) that, as the auctioned object be d e s i r a b l e to have i n Hui i s some measure - a homogeneous monetary l o a n , by which one can compare the 187 - it per- Table 5.1: Nash Bidding S t r a t e g i e s and T h e i r Derivatives R e s t r i c t i o n s : - no d e f a u l t , no l e a r n i n g - r i s k n e u t r a l i t y with p o s i t i v e time p r e f e r e n c e - l i n e a r d i s t r i b u t i o n of r e s e r v a t i o n d i s c o u n t v over - identical subjectively discounted expected First-Price Nash b i d d i n g s t r a t e g y , b v - Jfa ( v - v ) b 2 - 5 H v + Bid d i s t r i b u t i o n , F(b) (v-v)/(v-v) f s E(b ), E(b ) G - ± G* * v + ^ (v-v) - I s Slf (m+1) (m+2) H m = N-n+1: number of bidders at p e r i o d n - i G* G G* v - I * G* (v-v)/(v-v) v + — S i ! — (v-v) " (m+1) Var(b ) (v-v) 2 + s i r <*-2> " ¥ - I G* v - m (v-v) " 2 2 m . (v-v) (m+1) " - 1 G* 2 = y + f gains G* Second-Price 2 Var(b ), [v,v] m 2 + l . (m+1) (v-v) - _- G* 2 m 2 2(m-l)m (m+l) ^) 4 m ( - _ y ) 2 formance of d i f f e r e n t lending). This rates. One i s Hui from one's p o i n t measure would attempted to of preferably consider interest be the in (e.g., some rate r form that n borrowing or of interest solves equation below: (5.7) ^~J(A-b )(l+r ) k n _ k n + Z f k n n + 1 A(l+r ) ~ f s b . = 0 and b = b (b ) under the 0 n n n where k = NA+(N-n)(A-b ), n (5.7) fl first- (second-) p r i c e v auctioning method. This rowing measure i m p l i c i t l y assumes an i n d e n t i c a l and l e n d i n g — post interest rate for This internal rate of typical change Section between participant except for borrowing the out the like rate organizer bor- some k i n d of bidding a poor for at choice period given ex n. the and p a r t i c i p a n t N, solutions for r^. using and looks won turns y i e l d i n g non-unique 2, we m o t i v a t e d , Hui n who r e t u r n however flows which, signs twice, In tion Hui cash which m i s l e a d i n g l y interest a n u m e r i c a l example, lending rates. We now the distinc- formally define them. Definition 5.1: The ex post H u i borrowing i n t e r e s t won the l o a n at p e r i o d n ) , tal is the r a t e y b NA+(N-n)(A-b ) n where b n = b that solves equation n k (second-) p r i c e a u c t i o n i n g f = b, n k loan at period n ) , capi- (5.8) below: = ^!! t t f l A(l+Y b n i ) ~\ (5.8) s (b ), k = 1 n, under the first- method. D e f i n i t i o n 5.2: The ex post H u i l e n d i n g i n t e r e s t r a t e the to member n (who n = 1, . . . , N, with o p p o r t u n i t y c o s t of £^J(A-b )(l+r ) ~ = 0 and b, rate n = 1, to member n (who won N, with o p p o r t u n i t y - 189 - c o s t of capital r is n the r a t e v that n s o l v e s e q u a t i o n (5.9) ^:J(A-b )(l+Y^) n _ k k (second-) Note above that are the ex functions 2, rates the winning b i d s (or member's ex post to (5.9) under the first- the n t h member d e f i n e d the second h i g h e s t Hui i n t e r e s t where bids) rates will in at N- general program rate R b g i v e n by e x p r e s s i o n N-1, is Definition 2, 5.4: bidding e x a n t e H u i borrowing (5.10) below: the p r o b a b i l i t y that member i will b to b and E ( y ) (5.11): k 1 i n Table 5.1, S p e r i o d k by s u b m i t t i n g a b i d e q u a l n n = 1, (ex- o p p o r t u n i t y c o s t of c a p i t a l r^ and p l a n - 2 ^:J[A-E(b )][l+r ] " +^ k j o i n a H u i , an ex ante ( b ^ , b ^ > • • • »b^ JJ-1 ^ ' ^ k = 1, l o a n at solves equation is to perhaps more r e l e v a n t . Given member i ' s "^(b^), the d e c i d i n g on whether rate is 5.3: bidding interest ned . . . . n, Hui i n t e r e s t of one interest Definition for k n from a n o t h e r ' s . pected) win k = 1, post For an i n d i v i d u a l ned n + 1 n , and are independent of those at p e r i o d s n + l , Therefore, differ n A(l+r ) ~ , n p r i c e a u c t i o n i n g method. p e r i o d s 1, 1. = NA+(N-n)(A-b ) - ^ ! I f s b . = b „ = 0 and b. = b. (b, ) , 0 N k k k where below: b n + 1 N-1; E ( b ) is n and b Q n A[l+E(Y )] " k = NA+(N-n) [ A - E ( b ) ] n the expected effective (5.11) discount given = 0. Given member i ' s program o p p o r t u n i t y cost of c a p i t a l r^ and p l a n - ( b ^ , b ^ > • • • >b - 190 - ± N - 1 ) , his ex a n t e Hui lending Interest rate R i = is g i v e n by e x p r e s s i o n €\ tCl^-Wl where E ( Y ) s o l v e s e q u a t i o n n n k for n = 1, I 2, below: E K< ln> <tf' b 5 < ' 1 2 > (5.13): ^~J[A-E(b )][l+E(Y J)] " +^ k (5.12) n n + 1 A[l+r ] " k = NA+(N-n) [ A - E ( b ) ] . 1 H N - l ; E ( b ) and n b ( i ) n (5.13) fl a r e t n e same n a s n i- Definition 5.3. Note that the bids rates in Definitions those used his 5.3 in Definitions Essentially, is that are used and 5.4 5.1 are expected and 5.2 the ex ante Hui borrowing interest rates throughout the d u r a t i o n . induced are as well as time effective realized (lending) by the the interest of rates the expected preference 191 - interest discounts while discounts. r a t e to a member expected borrowing effective discounts d i f f e r e n t members to their expectations, characteristics, to d i f f e r e n t r e s e r v a t i o n d i s c o u n t s . - since ex ante effective In g e n e r a l , we would expect have d i f f e r e n t ex ante Hui i n t e r e s t rise calculate w i n n i n g - p r o b a b i l i t y - w e i g h t e d average (lending) streams, to may d i f f e r , income giving 6 AN APPLICATION TO COLLUSION AMONG SEVERAL SELLERS UNDER REPEATED AUCTIONS The rotating credit a p p l i e d to a form of quential via bidding sealed eligible that bid tacit with auctions, an sellers. a indivisible 1 2 previous sections agent buying commodity conspiracy is at regular from a in a be se- intervals, fixed non-cooperative can in group the of sense the i n d i v i d u a l behavior is i n the u s u a l Nash sense. Credit Collusion C o n s i d e r a buyer buying at sellers. in are agreed upon and f o l l o w e d , The Structure of Rotating single single indivisible The form of profit-maximizing studied c o l l u s i o n among a s m a l l group of s e l l e r s setting once the r u l e s 6.1 association 11 Actual commodity bids regular intervals supplied submitted by a s e l l e r are not to v i a sealed-bid auctions from a f i x e d exceed the buyer's a group of N reservation r price denoted by b . The s e l l e r s absence of any c o l l u s i o n , given with and the the the actions lowest spoils of cost be p r e - s e l e c t e d w i l l be s p l i t to win the the profits the a u c t i o n at In seller any p e r i o d a c c o r d i n g to some p r e - a g r e e d s h a r i n g r u l e . This 1 2 A s i m i l a r s t r u c t u r e can be d e f i n e d buyers repeated b i d d i n g s e t t i n g . is expected In a c o l l u s i o n with s i d e payments, L i section i n t o b i d d i n g at no c o s t s . s e l l e r s maximize t h e i r d i s c o u n t e d others. will enter essentially taken from Chew, Mao and Reynolds - 192 for - a single seller (1984). and several In r o t a t i n g c r e d i t collusion, the N s e l l e r s a g r e e , prior to the start 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 b i d l e v e l b which i s less r b . A t p e r i o d n d u r i n g an N - p e r i o d b i d d i n g c y c l e , o n l y the m (= N ' l i v i n g ' b i d d e r s (those who have yet to win an a u c t i o n i n any g i v e n than n+1) w cycle of 'dead' N periods) sellers are allowed (those who have to b i d at or won once p r i o r lower to the than b . The n - l given period) with- w draw by s u b m i t t i n g giving an appearnace centive after ner bid bids to do t h i s each of is to above the b , buyer of forfeiting still their chance being a c t i v e p r o v i d e d by the knowledge that to win but bidders. the game w i l l The i n restart the remaining l i v i n g s e l l e r s has won once. The f i n a l ' w i n w w i l l receive b a t p e r i o d N. T h i s i s accomplished by s u b m i t t i n g a w w of b u n d e r f i r s t - p r i c e a u c t i o n s or s u b m i t t i n g a b i d at l e s s than b w but receiving (e.g., 6.2 b f r o m the next h i g h e s t b i d submitted by a predetermined the winner at p e r i o d 1) dead s e l l e r under s e c o n d - p r i c e Assumptions The f o l l o w i n g n o t a t i o n w i l l be used i n t h i s N : the number of t>in : ith seller's b^ : the n b b s n n b b : the = sellers; b i d at p e r i o d n; lowest ( i . e . winning) b i d at p e r i o d n; second lowest b i d at p e r i o d n; f s b (b ) i n a f i r s t - p r i c e n n : the s i n g l e W section. (second-price) buyer's reservation : the withdrawal b i d l e v e l , b W price; r < b . - 193 - setting; auctions. Other n o t a t i o n s w i l l be e x p l a i n e d when they appear. Assumption 0: There i s no Assumption 1: no B a y i s i a n l e a r n i n g from past There i s Assumption C2: Each over time: default. seller c c^ = » ^2»* ** Assumption C3: Each s e l l e r profits n i has a d e t e r m i n i s t i c , c winning b i d s . and known c o s t s stream i t ' * ** ^ * maximizes h i s expected stream g i v e n a market d i s c o u n t factor We define below a r e s e r v a t i o n p r i c e v ^ based on the certain knowledge net value of his p. f o r the n present (Assumption 0) that ith seller the at p e r i o d worst he can do w is to r e c e i v e b at p-(»-")(v We w i l l r e f e r l n the Nth a u c t i o n : - C l n ) - b»-c to v? = {v. i 1 N . (6.1) ,...,v.„}, in n = 1, p r i c e of C4: Every [v L The seller by: seller the other s e l l e r s d i s t r i b u t i o n over ,v -n' at p e r i o d n-1 G, . i,n-l - G = °> r the at ith seller's O b v i o u s l y , v^ period n believes are drawn independently w = b for a l l that the i. reservation from a common u n i f o r m ]. n J d i s c o u n t e d expected i . . . . N, as — r e s e r v a t i o n p r i c e v e c t o r at p e r i o d n . Assumption 2, iN g a i n from f u t u r e p a r t i c i p a t i o n i n b i d d i n g f o r (n = 1, N-1), denoted by p{P, (b, )(b - v , ) + [ l - P , (b, ) ] G , }, in in n in in in in' 1 1 J > is defined (6.2) and where the P i,N-l i (b^ ) n auction n 6 denotes the i t h s e l l e r ' s with 2 < ' '> a b i d of b ^ n subjective at p e r i o d n . - 194 - p r o b a b i l i t y of winning The p r o b a b i l i t i e s of winning can be d e r i v e d a f t e r we o b t a i n the Nash e q u i l i b r i u m b i d d i n g s t r a t e g i e s the next s u b s e c t i o n . expected gain bidding cycle. Assumption In p a r t i c u l a r , G ^ Q r e p r e s e n t s from p a r t i c i p a t i o n C5: Every seller losing in the current f o r a l l i , where n = 1, the believes from p a r t i c i p a t i n g i n f u t u r e on in conspiracy that biddings the for a l l more a l i k e . other p e r i o d , are equal to h i s They are first the imposed i n order to who revealing is both property case of a seller of second-price 6 •1 and a b u y e r ) , second-price (Second-Price): second-price each expected gain sellers, own, contingent i.e., G. = G i n avoid using the tradi- auctions. Since auctions it is clear applies i n this (unlike that here. ro- the Hui the demand- (See Vickrey Hence, Under Assumptions 0 - 1 Nash e q u i l i b r i u m b i d d i n g program b. and C2 - for s e l l e r i C5, is the given by: b v.i n in + G. 'in f o r n = 1, N-l, (6.3) and b iN = V iN - e n making the m (= N-n+1) purely a s e l l e r (1961) and Cox, Roberson and Smith ( 1 9 8 2 ) . ) Theorem of Strategies t a t i o n c r e d i t c o l l u s i o n model, a b i d d e r i s bidder start assumption. Nash Equilibrium Bidding Consider the discounted . . . , N. t i o n a l homogeneous s e l l e r s 6.3 at discounted Assumptions C4 and C5 are symmetry assumptions sellers the o v e r a l l in < b w (6.3') - 195 - where f o r n = 1, G Proof: N-2, i,N-l and 6 = °« < ' 4 , ) Omitted. For the case s i v e l y defined b J in of first-price auctions, we p o s i t the following recur- b i d d i n g program: = v, + G. + ( l / m ) ( v - v , ) , in in n in for n - l , N-l, (6.5) and b V iN = iN = b W ( - The d i s t r i b u t i o n of b . i n , F. , is i given 6 V - ) by: n 1 - F . ( b . ) = (v - v , ) / ( v - v ) = n(v - b , +G, ) , in in n in n -n n in in v where given 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 is by P. (b. ) = [1-F, (b, ) ] in in in in m _ 1 . (6.6) J That (6.5) by showing is that participation expression ( 6 . 2 ) . Theorem 6.2 price it the maximizes current Nash the and discounted future demonstrated expected biddings, gain G^ _^> § i n from v e n by Hence, (First-Price): recursively Proof: in a Nash e q u i l i b r i u m b i d d i n g s t r a t e g y can be Under Assumptions equilibrium bidding by e x p r e s s i o n (6.5) p r o g r a m b^ with G^ Omitted. - 0-1 196 - and C2 - C5, the for the seller g i v e n by e x p r e s s i o n i first- is (6.4). given Note that the expected gains from p a r t i c i p a t i o n i n the b i d d i n g c o n s - p i r a c y G ^ are equal under f i r s t - p r i c e or s e c o n d - p r i c e a u c t i o n s . n ently, the per p e r i o d expected costs either auction i n s t i t u t i o n the Vickrey's remain light the of results case the for if we results are single-object extend of Cox, the from the b u y e r ' s same. T h i s appears auctions. model to Roberson and V i c k r e y ' s model. - 197 point - Smith of view under to correspond w i t h We do not incorporate Consequ- expect risk (1982)'s this to aversion in extension of 7 CONCLUSION This is gorous a first understanding mechanism for cations. giving perspective of 'rotational' b i d d i n g process as the To rate familarize several actual its uniqueness particularly strategy. rality reader competitive as w e l l with useful discount fairly for the the derivation borrowing and l e n d i n g of total ency surpluses members' measure interest interest allocations logical' for surplus, of joining rates (Little, shark or pawn shop, the the rotating determined criteria It by allo- began by the in- definition its existence turned out to be i n d i v i d u a l bidding we gave up a l o t as w e l l as the of gene- associated ex ante w i n - explicit and, a n a t u r a l candidate credit by not union, an almost b i d d i n g market lot, 1957), credit a rates. which p r o v i d e s different may be introduced a an o p t i m a l result, we illustrate and demonstrated A s i d e - r e w a r d from the Nash e x e r c i s e Is members' Hui to circumstances. at a r i - and savings Nash e q u i l i b r i u m b i d d i n g program and the ning-probability-weighted of loans Hui s t r u c t u r e , We then vector, general as small s t r u c t u r a l parameters. under obtain the examples and u s i n g In order to a c q u i r e t r a c t a b l e to post formation, the an a g e n t ' s r e s e r v a t i o n and ex from the r a t i o n a l c h o i c e interest t e r w o r k i n g of of attempt markets. seniority, forgetting and the For expression hence, f o r an insurance the effici- example, or by other the of loan 'socio- neighborhood companies.* loan 3 The We may a l s o examine a d i s c r i m i n a t i v e v e r s i o n of the r o t a t i o n a l b i d d i n g p r o c e s s where the w i n n e r 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 r a t h e r than A - b from b i d d e r j . n n - 198- total the of members' surplus also d u a l problem — the the membership. provides organizer's These are the natural objective problem — of promising topics the function for o p t i m a l combination for immediate follow-up research. Another d i r e c t i o n f o r f u t u r e work concerns the to rotational the to r e f l e c t Another Nash model aversion (e.g., von of the agents model is may have a large dimension loss in i n - o p t i m a l b i d d i n g s t r a t e g y may be the need to cover unexpected refinement s h a r i n g aspect of an i n s u r a n c e p r o b a b i l i t y of any one p e r i o d , and study how the modified risk We can i n t r o d u c e problem by adding an exogenous come at the bidding process. the r i s k to l o s s e s over introduce risk time-additive, Neumann-Morgenstern u t i l i t y time. aversion constant function with into relative their risk a v e r s i o n 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 i n t r o d u c i n g the p o s s i b i l i t y done, would theoretical ture of fault considerably explanation a rotating risk of lectively. and the enrich the behind the credit There appears pulates the to be a curious over the bers over change time the available, member's payoff preference, available rules risk as the should risk organizer provide sharing bears attitude, pool. default The o r g a n i z e r - 199 - of the to theory expli- i m p l i c i t l y mani- risk and r i s k i n e s s only h a l f de- to the i n f o r m a t i o n - the Hui o r g a n i z e r of structhe p a r a l l e l between agency share a to the membership c o l - structure relative while bearing and i f properly The p r i n c i p a l ' s problem i s and h i s membership such where but poses a common r i s k c i t l y manipulate the a g e n t ' s payoff technology model This, t y p i c a l l y observed r o t a t i o n a l b i d d i n g problem. monitoring current association each member, of d e f a u l t ? of by o p t i m i z i n g the can a l s o default other memattempt risk to instead. In the with opposite a monthly its 20 pool who are of a rotating credit approximately mutual friends, association $10,000, was completely the immediate financial credit horizon is the s e c t o r whose c a p i t a l i s associations reasonable Singapore, with approximation and other e s t i m a t e puts GNP the (Miracle, in the within the overall called be d e c l a r e d doing question by with the of the a lot acting Cohen, as 'ko' the in- of information of rota- and risk T h i s seems to be Taiwan, and many A f r i c a n Hong Kong, nations. One certain Japanese, are a description pocket the of the rotating sufficiently may its even pupolation credit associ- prevalent as to 1 4 "the market f o r the cause of lemons" conjecture the formal s e c t o r i n f i n a n c i n g efficiency may be (1970), that smaller, we have due to sector inherently the informal imperfect. sector i n i n f o r m a t i o n , m o n i t o r i n g , and even from conversations - 200 - a A complements by I t s rela- enforcement T h i s seems to f i n d at with under- informational shorter-term c a p i t a l , be r a t h e r u n o r t h o d o x ) . This i n f o r m a t i o n came student from H a w a i i . paper market f a i l u r e s the formal f i n a n c i a l unreasonable times, to Such In Hawaii today, among the more about at market 1980). countries, not (which may, capital Asian countries economy. illegal. greater the role f i n a n c e d mainly by numerous organizers to and developed asymmetry which makes tively away s i z e of E t h o p i a ' s i n f o r m a l f i n a n c i a l s e c t o r at 8% of Since A k e r l o f ' s stood east Miracle apply, ations, Vancouver, formed j o i n t l y a r b i t r a g e r s between the formal and the i n f o r m a l s e c t o r s . a in (Chang, 1981). Beyond ting loan members organizer formal direction, Japanese least American superficial financial tating support i n the c o e x i s t e n c e of the s o p h i s t i c a t e d institutions credit of the f o r m a l s e c t o r association f i n a n c i a l markets A s i a n i m m i g r a t i o n i n the developed 1980)) and other informal f i n a n c i a l i n s t i t u t i o n s , the above-mentioned In Section s m a l l group of is inversely setting fied 6, effects costs lers of have to his cost prior A different rotating in an costs the to of assumption. each default on horizon to that rotating the credit framework seller period and collusion collusion-with-bonding be adapted to shed l i g h t which are not draws analysis on t h i s of - 201 - the about a the selli- aversion. once a seller (1984), it symmetric analysis of whether has is there won i n a shown that Nash b e h a v i o r uncertain variable Radner (1980) Eswaran and Lewis question. from a necessarily with noncooperative The l e n g t h - o f - c o l l u s i o n auctions investigate c o l l u s i o n model i s under among a We may a l s o assume that agreement consistent such T h i s model may be m o d i - In Chew and Reynolds is Miracle i n a repeated every bidding the in the p r i v a t e but known and de- intertemporal r i s k related collusion infinite assume intertemporal u t i l i t i e s bidding period. credit of bidding s t r a t e g i e s . question an i n c e n t i v e Instead to and (Miracle, d i s t r i b u t i o n a l and i n f o r m a t i o n a l assumptions resulting additive particular g a i n from the we may San Diego; way i n which c o l l u s i o n w i t h minimal m o n i t o r i n g . streams, in countries the number of p a r t i c i p a n t s ) near to study the e f f e c t s is the poor' l o a n sharks and pawn shops. a simple i n a few ways. various on the (since place cost distribution we d e s c r i b e d related or extended terministic i n t r a - f a m i l y loans, sellers can take forms of very Indian centers as 'the (among b l a c k West in and Cohen, New York; ins- i n the West along with the r o - immigrants of Brooklyn, type financial and (1983) may a l s o REFERENCES A k e r l o f G. 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Smith, "Theory and Behavior of S i n g l e Object A u c t i o n s , " i n Research i n E x p e r i m e n t a l Economics V o l . 2 , e d . by V. L . Smith, Greenwich: JAI P r e s s , 1982. Cox, J . C , V . L . Smith, and J . M. W a l k e r , "Theory and Behavior of M u l t i ple Unit Auctions," Discussion Paper, Department of Economics, U n i v e r s i t y of A r i z o n a , 1982. D e l a n c e y , M. W . , " I n s t i t u t i o n s f o r the Accumulation and R e d i s t r i b u t i o n of Savings among M i g r a n t s , " J o u r n a l 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. E n g e l b r e c h t - W i g g a n s , R . , "Auctions and B i d d i n g Models: A Survey," ment S c i e n c e 26 (February 1980): 119-142. Manage- Eswaran, M. and T . R. L e w i s , " C o l l u s i v e Behaviour i n Repeated F i n i t e Games with B o n d i n g , " C a l t e c h S o c i a l Science Working Paper, 1983. F o r s y t h e , R. and R, M. I s a a c , "Demand R e v e a l i n g Mechanisms f o r P r i v a t e Good A u c t i o n s , " D i s c u s s i o n Paper, Department of Economics, U n i v e r s i t y of A r i z o n a , September 1980. G e e r t z , C , "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 : A ' M i d d l e Rung' i n Development," Economic Development and C u l t u r a l Change 10 ( A p r i l 1962): 241263. G r e e n , J . and J - J L a f f o n t , " C h a r a c t e r i z a t i o n of S a t i s f a c t o r y Mechanisms f o r the R e v e l a t i o n of P r e f e r e n c e s f o r P u b l i c Goods," Econometrica 45 (March 1977): 427-438. H a r r i s , M. and A . R a v i v , " A l l o c a t i o n Mechanisms and the Design t i o n s , " Econometrica 49 (November 1981): 1477-1499. Huang, Y . M . , p e r s o n a l Light, I. H . , Ethnic P r e s s , 1972. correspondence, Enterprise in of Auc- 1981. America, University of California L i t t l e , K . , "The Role of V o l u n t a r y A s s o c i a t i o n s i n West A f r i c a n U r b a n i z a t i o n , " American A n t h r o p o l o g i s t 59 (1957): 579-596. M i l g r o m , P. R . , "A Convergence Theorem f o r C o m p e t i t i v e B i d d i n g with f e r e n t i a l I n f o r m a t i o n , " Econometrica 47 (May 1979): 679-688. Dif- M i l g r o m , P . R. and R. J . Weber, "A Theory of A u c t i o n s and C o m p e t i t i v e d i n g , " Econometrica 50 (September 1982): 1089-1122. Bid- M i r a c l e , M. P . , D. S. M i r a c l e 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): 701724. Myerson, R . , "Optimal A u c t i o n D e s i g n , " 6 (1981): 58-73. Mathematics of O p e r a t i o n s Oren, M. E . and A . C . W i l l i a m s , Research 23 (November-December Oren, S. S. and M. H . Rothkopf, "Optimal B i d d i n g i n S e q u e n t i a l A u c t i o n s , " O p e r a t i o n s Research 23 (November-December 1975): 1080-1090. - "On C o m p e t i t i v e B i d d i n g , " 1975): 1072-1079. Research 203 - Operations Radner, R . , " C o l l u s i v e 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 , " J o u r n a l o f Economic Theory 22 (1980): 136-154. R i l e y , J . and W. Samuelson, "Optimal A u c t i o n s , " American Economic 71 (June 1981): 381-392. Review R o t h s c h i l d , M. and J . S t i g l i t z , "Equilibrium i n Competitive Insurance Markets: An Essay on the Economics of Imperfect Information," Q u a r t e r l y J o u r n a l of Economics (1976): 629-649. S t a r k , R. M. and M. H . Rothkopf, "Competitive B i d d i n g : A Comprehensive B i b l i o g r a p h y , " O p e r a t i o n s 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 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 among Urban Mexicans and C h i c a n o s , R u t g e r s , The State U n i v e r s i t y of New J e r s e y , 1983. V i c k r e y , W . , " C o u n t e r s p e c u l a t i o n , A u c t i o n s and C o m p e t i t i v e d e r s , " J o u r n a l of F i n a n c e 16 (May 1961): 8-37. W i l s o n , R . , "A B i d d i n g Model of P e r f e c t S t u d i e s (October 1977): 511-518. , Sealed C o m p e t i t i o n , " Review of Economic "Competitive Exchange," Econometrica 46 (May 1978): , "Auctions of S h a r e s , " Q u a r t e r l y J o u r n a l o f Economics 1979): 675-689. - 204 - Ten- 577-585. (November APPENDIX P r o o f o f Lemma 3 . 1 : By the d e f i n i t i o n of v ^ , V'l.N-l* - U Y i< iN> V - where Y i,N-l = ^iO^'^r^^il^-'^^.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 . . ) i n c r e a i ' i i,N-l i,N-l I iN J ses T J in Given m o n o t o n i c i t y and p o s i t i v e V ^ V -i,N-l where ^ V i,N-l i.N-1' v ^ ^ = - |I ^ - I ^ ^ | and v ^ ^ = * V^.N-lK.N-l^i.N-l*' = U Y v 3 = 5 ) *• " l < i H l i H - l " 2 ± , N - l » f Again, positive and = (N+1)A. Let V^.N-lk.N-l^i.N-l*' V^NKN-I^.N-I^ time p r e f e r e n c e and m o n o t o n i c i t y imply that a > d, (A.l) a > a, (A.2) d < d, (A.3) a < d. (A.4) Since (A.4) time p r e f e r e n c e , we have i s assumed c o n t i n u o u s , inequalities (A.l), (A.2), ( A . 3 ) and imply that v^ ^ ^ e x i s t s and i s u n i q u e . By s i m i l a r arguments, i t can be e s t a b l i s h e d - 205 - that, in general, there e x i s t s a unique v . = ( v , v , v . „ , ) with v . e [ v . , v . ] , where i i l 12 ' i,N-l in -in' in J v. -in = min{l, . , - 1 , , T?—— [ I . - I , ,.+(N-n-l)v. ,.]}, i,n+l i n N-n i n i , n + l i,n+l " 1 J N+1 . A, N-n+1 in for 1 a l l n £ {l,...,N-l}, such that e q u a t i o n ( 3 . 1 ) h o l d s . Q . E . D . Proof of Theorem 4.1: At period utility, K n, b^, k = 1 ^€i ik x x u i i< ik- choosing ( A v - ik 0, b. in 1, + \ u J i ( i + i N J A - ( N -j ) v i ) + J u (I 2, 3' and 4, the problem of member i at p e r i o d n EU i s e q u i v a l e n t to maximizing n b E u f = J A., in in (b, ) in U ^in i " T~~ in ( I in + N A N n V -( - ) in> + E X U k!n+l ik i ( I ik- A ) HF i n N {u (I. +NA-(N-n)b )-u (I +NA-(N-n)v )}[F (b )] i C n in (b N l n )] - ^in-i^in-^-in^K^Hl-tF^Cb^)]^}, " TT~ in + Note r in< inKl = "XT" in = A) Cj+i ik i ik- ' to maximize E u f ( b . ) given i n expression ( 4 . 3 ) . in in n b ) (A.5) t h a t maximizing E U ^ ( b £ ) in< in> ) N. Under assumptions E Given t i m e - a d d i t i v e the c o n d i t i o n a l r e s e r v a t i o n u t i l i t y of member i at p e r i o d n i s j = n , n+1, is n - l , a r e known. i n i i n ^i^in-^-i^in-^in) l n + in i n n i n ^ + 1 ^ I n ^ * > (A.6) where - 206 - n E ?i , n +j.il = i4, n +.i_i( l i* ,n+l E The f i r s t suppressing = Max E.i , n + l..(b.i , n + l..)• b order c o n d i t i o n f o r b ^ the to maximize e x p r e s s i o n n i n d i v i d u a l and time s u b s c r i p t s , N n 1 (A.6), after is , , 0 = E'(b) = -(N-n)F(b) " ~ {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 * = Note that (A.7) With b. ( ji , n. +i l / Ki n ) *i , nJ.I +l • E implies increasing With d e c r e a s i n g equation (4.5) is (4.5) is equation utility, (4.5). the RHS of equation (4.5) is increasing marginal-outbidden-rate d i s t r i b u t i o n F(b), decreasing in b. Therefore, the bid the that in LHS of satisfies unique. The second order c o n d i t i o n f o r b to maximize ( A . 6 ) r e q u i r e s (N-n)u"(I+NA-(N-n)b)F(b) - that [(N-n+l)u*(I+NA-(N-n)b)+u'(I-A+b)]F'(b) + <t>F"(b) < 0, which i s satisfied decreasing is marginal l o c a l l y by the b that outbidden r a t e . solves equation Hence, the b i d that (4.5) if F yields maximizes (A.6) unique. Q.E.D. - 207 - Proof of Corollary 4.5; ob* We want to show that the sign of is d e r i v a t i v e of both s i d e s of e q u a t i o n ( A . 7 ) 0 = E"(b*)(ob*) 5 + E ' ( b * ) to positive. Take total obtain (ov) ov N n 1 = E"(b*)(ab*) + ( N - n ) [ F ( b * ) ] ~ ~ F ' ( b ) ( | i ) ( 5 v ) . Therefore | £ = -{(N-^tFCb^l^-S'Cb^^l/E^Cb*), where = (N-n)u'(I+NA-(N-n)v) + u'(I-A+v) > 0. ov 9b* Since E"(b*) < 0 and F ' ( b ) The > 0, we have p r o o f f o r the s e c o n d - p r i c e Proof of Corollary > 0 as case i s desired. similar. Q.E.D. 4.7; Take the t o t a l d e r i v a t i v e of (A.7) to obtain dE! (b* ) 0 = E" (b* )(3b* ) + [ — " —](9X, ) in in in oX in in X / v which L J implies 9b* in 9E*. (b* ) . m in' 1 oX. E" (b* )' in in in' v ax. in Since 9E'. (b* ) i n i n = ( N - n ) [ F , (b* ) ] ^ i^n m ax, in L and E ' ^ ( b * ) < 0, n The n v N h\ (b* ) - E * . • T4^2 in i_n ' i.,.n. .+. .l . (X ), * ' in n J > °» y we have shown that proof of the s e c o n d - p r i c e a D * n / case i s a ^ n > 0 < similar. Q.E.D. - 208 - Proof of Theorem 5 » 1 ; In t h i s proof, without n to s i m p l i f y f e a r of c o n f u s i o n , we w i l l omit the time index the n o t a t i o n . Suppose, a t p e r i o d n, b i d d e r j b e l i e v e s that a l l h i s r i v a l s adopt the bidding strategy function (5.3). uniformly distributed over [v,v]. By assumption 4', v^, f o r a l l i * j , i s Since b^ i s l i n e a r i n v^, by Assumption 4', b^ w i l l be u n i f o r m l y d i s t r i b u t e d over [b,b], where -i- b = v G*, — m i b - v - (v-v) - - G*. m i - ^ r - m+1 Recall the t h a t m = N-n+1 i s the number of b i d d e r s at p e r i o d n and that GJ i s identical s u b j e c t i v e l y discounted expected gain from future bidding participation. The i n v e r s e of (5.3) i s v, = f (b, ) = — i i m [b - - L _ v + - G*] . • i m+1 m x Therefore, f(b F,(b.) = F (v.) = / b l v i v J ) v-v d[ ] = n [b, — v + — G*], i m i v-v L and F^(b ) = n = ± (^t!)/(v-v). Under Assumptions 0, 1, 2, 3", 4' and 5, the expected utility differ- ence f o r bidder j of a b i d b . i s E.(b.) 3 J = ( m - l ) ( v -b ) [ n ( b -v+ I J J J G*)] J m _ 1 + (m-l)j£ (x-v +G*)[n(x-y+ I j J J - 209 - m 2 G*] " ndx, J = (m-l)Ti m 1 { ( v . - b . ) ( b -v+ - G * ) j J J m 2 m 1 The f i r s t ( x - v . + G * ) ( x - v + I G*) J J m j + 1 order c o n d i t i o n f o r b*. to maximize E . J J m 2 ~ dxV. ( A . 9 ) is 0 = E'.(b*) 3 J = ( m - l ) T i " ( b * - v + I G * ) ~ ' [ - ( b -v+ - G*)+(m-l)(v . - b . ) - ( b - v .+G*) ] 2 ~ m 3 J ~ m J J J J J J m 1 m = (m-l)r| " (b*-v+ i m 1 m 2 2 G*) " [-(m+l)b*+mv.+v- — G*] , (A.10) which i m p l i e s b* = 2 It is (mv. + v - — j m m+1 G*) = v . - - ^ p ( v . - v ) 2 J m+1 J " s t r a i g h t f o r w a r d to check 3 since bidder j's shown that bidding 3 Thus b* a l s o J optimal strategy the strategy It 1 3 m > 1. then strategy ~ m satisfies (A.ll) is function m G*] ~ J by < 0 the same as a l l h i s r i v a l s ' , (5.3) is indeed a Nash Since we have equalibrium at p e r i o d n . follows (5.3), 2 the second order c o n d i t i o n . that the v e c t o r b. = ( b . . , b . _ , . . . , b . i given (A.ll) that E"(b*) = - ( m - l ) ( m + l ) T i " [ b * - v + m - - G*. m j where n = 1, 2, i l ' i2' N-1, is , ), ' i,N-l ' a Nash e q u i l i b r i u m with b. in bidding program. Q.E.D. - 210 -
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Two essays on financial economics : I. Weighted utility, risk aversion and portfolio choice : II. Competitive.. Mao, Mei Hui Jennifer 1985-12-31
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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 | 1985 |
Date Issued | 2010-06-23T19:55:16Z |
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 |
Date Available | 2010-06-23 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0096730 |
URI | http://hdl.handle.net/2429/25936 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
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