Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Two essays on financial economics : I. Weighted utility, risk aversion and portfolio choice : II. Competitive.. Mao, Mei Hui Jennifer 1985-12-31

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
[if-you-see-this-DO-NOT-CLICK]
UBC_1985_A1 M36.pdf [ 8.73MB ]
Metadata
JSON: 1.0096730.json
JSON-LD: 1.0096730+ld.json
RDF/XML (Pretty): 1.0096730.xml
RDF/JSON: 1.0096730+rdf.json
Turtle: 1.0096730+rdf-turtle.txt
N-Triples: 1.0096730+rdf-ntriples.txt
Original Record: 1.0096730 +original-record.json
Full Text
1.0096730.txt
Citation
1.0096730.ris

Full Text

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.  No matter  i n which of  the  above  they should h e l p us understand  vioral  proved  results  This  between r i s k  f u n c t i o n depends on the u n d e r l y i n g d i s t r i b u t i o n ,  function  be,  equivalent,  risk  logy of  The e q u i v a l e n c e  spread and r i s k a v e r s i o n  introducing  modification  necessary  demand i s  the  certainty  Most of  hypotheses.  might  categories  the nature  implications.  -  123  -  of  the  findings  t u r n out  to  the  r e l a t e d market beha-  REFERENCES  A l l a i s , M . , "Le Comportement de l'Homme R a t i o n n e l Devant l e R i s q u e , C r i t i que des P o s t u l a t s et Asiomes de l ' E c o l e A m e r i c a i n e , " Econometrica 21 (1953): 503-546. —  , "The S o - C a l l e d A l l a i s Paradox and R a t i o n a l D e c i s i o n s under U n c e r t a i n t y , " i n Expected U t i l i t y Hypotheses and the A l l a i s Paradox e d . by M. A l l a i s and 0. Hagen, H o l l a n d : D. R e i d e l , 1979, 437-682.  Anscombe, F . J . and R. J . Aumann, "A D e f i n i t i o n of S u b j e c t i v e P r o b a b i l i t y , " Annals of Mathematical S t a t i s t i c s 34 (1963): 199-205. Arrow, K . 1971.  J.,  Essays  in  the  , "Risk P e r c e p t i o n I n q u i r y 20 (1982): 1-9. Bell,  Theory  in  of  Risk  Bearing,  Psychology  D. E . , "Regret i n D e c i s i o n Research 30 (1982): 961-981.  Making under  and  Chicago:  Economics,"  Uncertainty,"  B o r c h , K . , "A Note on U n c e r t a i n t y and I n d i f f e r e n c e Economic S t u d i e s (January 1969): 1-4.  Markham,  Curves,"  Economic  Operations  Review  of  Cass,  D. and J . E . S t i g l i t z , "The S t r u c t u r e of I n v e s t o r P r e f e r e n c e s and A s s e t R e t u r n s , and S e p a r a b i l i t y i n P o r t f o l i o A l l o c a t i o n : A C o n t r i b u t i o n to the Pure Theory of Mutual F u n d s , " J o u r n a l of Economic Theory 2 (1970): 122-160.  Cass,  D. and J . E . S t i g l i t z , "Risk A v e r s i o n and Wealth E f f e c t s on P o r t f o l i o s with Many A s s e t s , " Review of Economic S t u d i e s ( J u l y 1972): 331354.  Chew S. H . , "Two R e p r e s e n t a t i o n Theorems and T h e i r A p p l i c a t i o n to T h e o r y , " P h . D . D i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia,  Decision 1980.  , "A M i x t u r e Set A x i o m a t i z a t i o n of Weighted U t i l i t y T h e o r y , " Economics Department Working Paper, U n i v e r s i t y of A r i z o n a , 1981. , "Weighted U t i l i t y Theory, C e r t a i n t y E q u i v a l e n c e , and General Monetary L o t t e r i e s , " Economics Department Working Paper, U n i v e r s i t y of A r i z o n a , J u l y 1982. , "A G e n e r a l i z a t i o n of the Q u a s i l i n e a r Mean w i t h A p p l i c a t i o n s to the Measurement of Income I n e q u a l i t y and D e c i s i o n s Theory R e s o l v i n g the A l l a i s Paradox," Econometrica 51 ( J u l y 1983): 1065-1092.  -  124  -  , "Between Strong S u b s t i t u t i o n and Very Weak S u b s t i t u t i o n : Imp l i c i t Weighted U t i l i t y and Semi-Weighted U t i l i t y T h e o r y , " Economics Working Paper, Johns Hopkins U n i v e r s i t y , 1984, Revised 1985. ——, "A Note on Working Paper, 1985.  Differentiability  of  Preference  Functionals,"  Chew S. H . and K . R. MacCrimmon, "Alpha U t i l i t y Theory: A G e n e r a l i z a t i o n of Expected U t i l i t y T h e o r y , " U n i v e r s i t y of B r i t i s h Columbia Working Paper 669, 1979a. , "Alpha U t i l i t y Theory, L o t t e r y Compositions and the A l l a i s Paradox," U n i v e r s i t y of B r i t i s h Columbia Working Paper 686, 1979b. Cox,  J . C , B. Roberson and V . L . 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, ed.. by V . L . Smith, V o l . 2 , Greenwich: J A I , 1982.  DeGroot, M. H . , O p t i m a l S t a t i s t i c a l  Decision,  D e k e l , E . , "Asset Demands Without the s i t y m a n u s c r i p t , February 1984.  New York: M a C r a w - H i l l ,  1970.  Independence Axiom," Harvard U n i v e r -  Diamond, P. A . and J . E . S t i g l i t z , "Increases i n R i s k and i n R i s k A v e r s i o n , " J o u r n a l of Economic Theory 8 (1974): 337-360. Dybvig, P. H. , "Rocovering A d d i t i v e U t i l i t y Economic Review 24 (June 1983): 379-396.  Functions,"  Edwards, W . , "The Theory (1954): 380-417.  Psychological  of  Decision  Making,"  E i s n e r , R. and R. H . S t r o t z , " F l i g h t Insurance and the J o u r n a l of P o l i t i c a l Economy 69 (1961): 355-368.  International  Bulletin  Theory of  51  Choice,"  E p s t e i n , L . G . , "Decreasing R i s k A v e r s i o n and Mean-Variance A n a l y s i s , " Economics Working Paper 8406, U n i v e r s i t y of T o r o n t o , January 1984. F e l d s t e i n , M. S . , "Mean-Variance A n a l y s i s i n the Theory of L i q u i d i t y P r e f e r e n c e and P o r t f o l i o S e l e c t i o n , " Review of Economic S t u d i e s (January 1969): 5-12. F i s h b u r n , P. C , 1970.  U t i l i t y Theory f o r D e c i s i o n Making, New York: John W i l e y ,  , " N o n t r a n s i t i v e Measurable Psychology (August 1982): 31-67. , " T r a n s i t i v e Measurable (December 1983): 293-317.  -  Utility,"  Utility,"  125  -  Journal  Journal  of  of  Mathematical  Economic  Theory  Friedman, M. and L . J . Savage, "The U t i l i t y A n a l y s i s of Choices R i s k , " J o u r n a l of P o l i t i c a l Economy 56 (1948): 279-304. G r e t h e r , D. and C . P l o t t , R e v e r s a l Phenomenon," 638.  Involving  "Economic Theory of Choice and the Preference American Economic Review 69 (August 1979): 623-  Handa, J . , " R i s k , P r o b a b i l i t i e s and a New Theory of C a r d i n a l J o u r n a l of P o l i t i c a l Economy (February 1977): 97-122. Hanoch, G. and H . Levy, " E f f i c i e n c y A n a l y s i s of Choices Review of Economic S t u d i e s 36 (1969): 335-346. Hardy, G . H . , J . E . Cambridge, 1934. Hart,  Littlewood  and  G.  Poyla,  Utility,"  Involving  Inequality,  Risk,"  Cambridge:  0. D . , "Some Negative R e s u l t s on the E x i s t e n c e of Comparative S t a tics Results in Portfolio T h e o r y , " Review of Economic Studies (October 1975): 615-621.  H e r s t e i n , I . N. and J . M i l n o r , "An A x i o m a t i c Approach to Measurable t y , " Econometrica 21 ( A p r i l 1953): 291-297. Kahneman, D. and A . T v e r s k y , "Prospect Theory: An A n a l y s i s under R i s k , " Econometrica 47 (March 1979): 263-291.  of  Utili-  Decision  Karmarkar, U . S . , " S u b j e c t i v e l y Weighted U t i l i t y : A D e s c r i p t i v e E x t e n s i o n of the Expected U t i l i t y M o d e l , " O r g a n i z a t i o n a l Behavior and Human Performance 21 (1978): 61-72. Kihlstrom, R. E . and Mirman, "Risk A v e r s i o n w i t h J o u r n a l of Economic Theory 8 (1974): 361-388.  Many  Commodities,"  K i h l s t r o m , R. E . , D. Romer, and S. W i l l i a m s , "Risk A v e r s i o n with Random I n i t i a l W e a l t h , " Econometrica 49 ( J u l y 1981): 911-920. K u n r e u t h e r , H . , R. G i n s b e r g , L . M i l l e r , P. S a g i , P . S l o v i c , B . B o r k a n , and N . K a t z , D i s a s t e r Insurance P r o t e c t i o n : P u b l i c P o l i c y L e s s o n s , New York: W i l e y , 1978. L e l a n d , H . E . , "Saving and U n c e r t a i n t y : The P r e c a u t i o n a r y Demand S a v i n g , " Q u a r t e r l y J o u r n a l of Economics 82 (1968): 465-473. Levy,  for  H . and Y . K r o l l , "Investment D e c i s i o n R u l e s , D i v e r s i f i c a t i o n , and the I n v e s t o r ' s I n i t i a l W e a l t h , " Econometrica 46 (September 1978): 1231-1237.  L i c h t e n s t e i n , S. and P . S l o v i c , "Reversal of Preference between Bids and Choices i n Gambling D e c i s i o n s , " J o u r n a l o f E x p e r i m e n t a l Psychology 89 (1971): 46-55.  -  126  -  L i n t n e r , J . , "The V a l u a t i o n of R i s k y A s s e t s and the S e l e c t i o n of R i s k y Investments i n Stock P o r t f o l i o s and C a p i t a l Budgets," The Review of Economics and S t a t i s t i c s 47 (February 1965): 13-37. Loomes, G. and R. Sugden, "Regret Theory Economics L e t t e r s 12 (1983): 19-21. Luenberger, Wiley,  D. G . , O p t i m i z a t i o n by V e c t o r 1969.  and  Space  Measurable  Methods,  Utility,"  New York:  John  MacCrimmon, K . R. and S. L a r s s o n , " U t i l i t y Theory: Axioms v e r s u s 'Paradoxes'," i n Expected U t i l i t y Hypotheses and the A l l a i s Paradox, e d . by M. A l l a i s and 0. Hagen, H o l l a n d : D. R i e d e l , 1979, 333-409. M a c h i n a , M. J . , "'Expected U t i l i t y ' A n a l y s i s without Axiom," Econometrica 50 (March 1982a): 277-323. , "A Stronger C h a r a c t e r i z a t i o n of Econometrica 50 ( J u l y 1982b): 1069-1079.  the  Declining  Marschak, J . , "Rational Behavior, Uncertain Prospects, U t i l i t y , " Econometrica 18 (1950): 111-141. Markowitz, H . , " P o r t f o l i o , "The U t i l i t y (1952b): 151-158.  Selection," of  J o u r n a l of Finance  Wealth,"  Journal  , Portfolio Selection: Efficient New York: John W i l e y , 1959.  of  Independence  Risk  Aversion,"  and  Measurable  (1952a):  Political  Diversification  of  Economy  J . R . , " A l t e r n a t i v e s to the Expected U t i l i t y R u l e , " d i s s e r t a t i o n , U n i v e r s i t y of C h i c a g o , 1977.  and  Their  unpublished  Menezes, C . F . and D. L . Hanson, "On the Theory of R i s k A v e r s i o n , " n a t i o n a l Economic Review 11 (October 1970): 481-487. Nakamura, Y . , "Nonlinear U t i l i t y A n a l y s i s , " P h . D . D i s s e r t a t i o n , of C a l i f o r n i a at D a v i s , 1984.  60  Investments ,  Mayshar, J . , "Further Remarks on Measures of R i s k A v e r s i o n U s e s , " J o u r n a l of Economic Theory 10 (1975): 100-109. Meginniss, Ph.D.  77-99.  Inter-  University  Nashed, M. Z . , " D i f f e r e n t i a b i l i t y and R e l a t e d P r o p e r t i e s of N o n l i n e a r Oper a t o r s : Some Aspects of the Role of D i f f e r e n t i a l s i n N o n l i n e a r F u n c t i o n a l A n a l y s i s , " i n N o n l i n e a r F u n c t i o n a l A n a l y s i s and A p p l i c a t i o n s , ed. by L . B. R a i l , New York: Academic P r e s s , 1971, 103-310. von Neumann, J . and 0. Morgenstern, Theory of Games and Economic B e h a v i o r , P r i n c e t o n : P r i n c e t o n U n i v e r s i t y P r e s s , 2nd e d i t i o n , 1947.  -  127  -  Paroush, J . , "Risk Premium with Theory 11 (1975): 283-286.  Many Commodities,"  Journal  of  Economic  P a s h i g i a n , B . P . , L . L . Schkade and G. H . Menefee, "The S e l e c t i o n of An Optimal D e d u c t i b l e f o r A Given Insurance P o l i c y , " J o u r n a l of B u s i n e s s 39 (1966): 35-44. Payne, J . W . , " A l t e r n a t i v e Approaches to D e c i s i o n Making under R i s k : Moments versus R i s k Dimensions," P s y c h o l o g i c a l B u l l e t i n 80 (1973): 439453. P o l l a k , R. A . , "The R i s k 1973): 35-39.  Independence  P r a t t , J . W . , "Risk A v e r s i o n 32 (1964): 122-137.  Axiom," Econometrica  41  (January  i n the Small and i n the L a r g e , " Econometrica  P r a t t , J . W., H . R a i f f a , and R. S c h l a i f e r , "The Foundations of D e c i s i o n s under U n c e r t a i n t y : An Elementary E x p o s i t i o n , " J o u r n a l o f the American S t a t i s t i c a l A s s o c i a t i o n 59 (1964): 353-375. Ramsey, F . D . , "Truth and P r o b a b i l i t y , " Mathematics, e d . by R. B . B r a i t h w a i t e , Ross,  (1926) i n The Foundations Humanities P r e s s , 1931.  of  S. A . , "Some Stronger Measures of R i s k A v e r s i o n i n the Small and the Large with A p p l i c a t i o n s , " Econometrica 49 (May 1981): 621-638.  R o t h s c h i l d , M. and J . E . S t i g l i t z , "Increasing Risk J o u r n a l of Economic Theory 2 (1970): 225-243. , "Increasing Risk I I : Its Economic Theory 3 (1971): 66-84.  Economic  Samuelson, P. A . , " P r o b a b i l i t y , Utility Econometrica 20 (1952): 670-678.  and  Sandmo, A . , "Capital Risk, Consumption, Econometrica (October 1969): 586-599.  I:  A  Definition,"  Consequences,"  J o u r n a l of  the  Independence  and  Portfolio  Choice,"  Decisions,"  Review of  , "The E f f e c t of U n c e r t a i n t y on Saving Economic S t u d i e s ( J u l y 1970): 353-360.  Axiom,"  , "Portfolio Theory, A s s e t Demand and T a x a t i o n : Comparative S t a t i c s with Many A s s e t s , " Review o f Economic S t u d i e s (June 1977): 369-379. Savage, L . J . , The Foundations  of S t a t i s t i c s ,  New York: John W i l e y ,  1954.  S e l d e n , L . , "A New R e p r e s e n t a t i o n of P r e f e r e n c e over ' C e r t a i n x U n c e r t a i n ' Consumption P a i r s : The ' O r d i n a l C e r t a i n t y E q u i v a l e n t ' Hypothesis," Econometrica 46 (September 1978): 1045-1060.  -  128  -  Sharpe, W. F . , " C a p i t a l Asset P r i c e s : A Theory of Market E q u i l i b r i u m under C o n d i t i o n s of R i s k , " J o u r n a l o f F i n a n c e 19 (September 1964): 425-442. S h e f r i n , H . M. and M. Statman, " E x p l a i n i n g I n v e s t o r P r e f e r e n c e f o r Cash D i v i d e n d s , " J o u r n a l of F i n a n c i a l Economics 13 (1984): 253-282. S l o v i c , P . , B. F i s c h h o f f , S. L i c h t e n s t e i n , B. C o r r i g a n , and B . Coombs, "Preference f o r I n s u r i n g a g a i n s t Probable Small L o s s e s : Insurance I m p l i c a t i o n s , " J o u r n a l of R i s k and Insurance 44 (1977): 237-258. S t i g l i t z , J . E . , "A Consumption-Oriented Theory of the Demand f o r F i n a n c i a l A s s e t s and the Term S t r u c t u r e of I n t e r e s t R a t e s , " Review of Economic S t u d i e s ( J u l y 1970): 321-351. , " E f f e c t s of Wealth, Income and C a p i t a l Gains T a x a t i o n on R i s k T a k i n g , " Q u a r t e r l y J o u r n a l of Economics (May 1969): 263-283. Tetens, J . N . , E i n l e i t u n g t e n , L e i p z i g (1789).  zur Berechnung  der L e i b r e n t e n  und Anwartschaf-  T h a l e r , R . , "Toward A P o s i t i v e Theory of Consumer C h o i c e , " Economic B e h a v i o r and O r g a n i z a t i o n 1 (1980) : 39-60.  Journal  of  T h a l e r , R. and S h e f r i n , "An Economic Theory of P o l i t i c a l Economy 89 (1981): 392-410.  Self-Control,"  Journal  of  Tobin, J . , " L i q u i d i t y Preference as Behavior Economic S t u d i e s 25 (1958): 65-86.  Towards  Review  of  Risk,"  T r e y n o r , J . L . , "Toward A Theory of Market Value of R i s k y A s s e t s , " Unpub l i s h e d m a n u s c r i p t , 1961. T v e r s k y , A . and D. Kahneman, "Judgement under U n c e r t a i n t y : B i a s e s , " S c i e n c e 185 (1974): 1124-1131. , "The Framing of D e c i s i o n s S c i e n c e 211 (1981): 453-458.  and  the  Heuristics  Psychology  of  and  Choice,"  Weber, R. J . , "The A l l a i s Paradox, Dutch A u c t i o n s , and A l p h a - U t i l i t y T h e o r y , " Northwestern U n i v e r s i t y Working Paper, September 1982. Y a a r i , M. E . , "Convexity J o u r n a l o f Economics  i n the Theory of Choice 79 (1965): 278-290.  under  Risk,"  Quarterly  , "Some Remarks on Measures of R i s k A v e r s i o n and on T h e i r Uses," J o u r n a l of Economics Theory 1 (1969): 315-329. Zeckhauser, R. and E . K e e l e r , " A n o t h e r Type Econometrica 38 (September 1970): 661-665.  -  129  -  of  Risk  Aversion,"  ESSAY  COMPETITIVE  BIDDING  AND  INTEREST  RATE  I I  FORMATION  IN  AN  -  130  -  INFORMAL  FINANCIAL  MARKET  0 INTRODUCTION  One  problem with  scarcity,  which  busi-nessmen taneously  these  informal  which,  i n the  as  'rotating  the tual  economy  bank  wide-spread  in  make r e g u l a r  borrowing needs organization,  for  country i s  inaccessible  As a r e s u l t ,  world. are  a broad c l a s s of  spon-  literature,  association'. into  individual  the  are most  members  which  cultural  conditions.  (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. A . , "The Market f o r 'Lemmons': Q u a l i t y U n c e r t a i n t y and the Mark e t Mechanism," Q u a r t e r l y J o u r n a l of Economics 84 (August 1970): 488500. A r d e n e r , S. G . , "The Comparative Study 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 , " J o u r n a l of the R o y a l A n t h r o p o l o g i c a l I n s t i t u t e of Great B r i t a i n and I r e l a n d 94, Part II (1964): 201-229. Bonnett, A . W., I n s t i t u t i o n a l A m e r i c a : An A n a l y s i s of Press of A m e r i c a , 1981.  A d a p t a t i o n of West I n d i a n Immigrants to Rotating Credit Associations, University  Cassady, R . , A u c t i o n s and A u c t i o n e e r i n g , Berkeley f o r n i a : U n i v e r s i t y of C a l i f o r n i a P r e s s , 1967. Chang,  P. L . , p e r s o n a l  Chao Ming, 11.  "A Report  communication,  and Los A n g e l e s ,  Cali-  1981.  on Hui D e f a u l t s , "  Chao Ming Monthly (June  1983):  1-  Chew S. H . , Mao M. H . and S. S. Reynolds, "Rotating C r e d i t C o l l u s i o n i n Repeated A u c t i o n s with a S i n g l e Buyer and S e v e r a l S e l l e r s , " Economics L e t t e r s 16 (1984): 1-6. Chew S. H . and S. S. Reynolds, "A R o t a t i n g C r e d i t S t r u c t u r e f o r T a c i t C o l l u s i o n i n Repeated A u c t i o n s with I n f i n i t e H o r i z o n , " U n i v e r s i t y of A r i z o n a Working Paper, 1983. Chua,  B. H . , "The C h i t Fund i n Singapore: F a i l u r e of A L a r g e - S c a l e 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 , " i n Southeast A s i a : Women, Changing S o c i a l Structure and C u l t u r a l Continuity, edited by G. B. H a i n s w o r t h , U n i v e r s i t y of Ottawa P r e s s , 1981, 119-134.  C o p p i n g e r , V . , V . L . Smith, and J . T i t u s , " I n c e n t i v e s and English, Dutch, and Sealed-Bid Auctions," Economic (January 1980): 1-22.  Behavior Inquiry  in 18  Cox,  J . C , B. Roberson and V . L . 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  -  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Country Views Downloads
United States 22 2
China 15 25
Russia 12 0
Japan 10 0
Thailand 2 0
Germany 2 5
Poland 2 0
Brazil 1 0
France 1 0
Philippines 1 0
City Views Downloads
Unknown 21 5
Fort Worth 10 0
Tokyo 10 0
Beijing 10 0
Shenzhen 5 25
Ashburn 4 0
Saint Petersburg 3 0
Phoenix 2 0
Mountain View 1 1
Buffalo 1 0
Wilmington 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0096730/manifest

Comment

Related Items