nicholson微观经济理论（答案+ppt+习题库）-80-03-033593-0

EXPECTEDUTILITYANDRISKAVERSIONINTHISCHAPTERWELOOKATSOMEOFTHEBASICELEMENTSTHATCHARACTERIZEINDIVIDUALSMOTIVATIONWHENMAKINGCHOICESINUNCERTAINSITUATIONSWESHOWHOWTHENOTIONOFUTILITYCANBEGENERALIZEDTOAPPLYTOCASESINWHICHOUTCOMESARESUBJECTTOSOMEDEGREEOFRANDOMNESSWETHENUSETHISEXPANDEDCONCEPTOFUTILITYTOEXAMINETHEPHENOMENONOF“RISKAVERSION”THATIS,WESHOWWHYINDIVIDUALSGENERALLYDISLIKEUNCERTAINSITUATIONSANDMAYBEWILLINGTOPAYSOMETHINGTOREDUCETHEUNCERTAINTYTHEYFACE8CHAPTERPROBABILITYANDEXPECTEDVALUETHESTUDYOFINDIVIDUALBEHAVIORUNDERUNCERTAINTYANDTHEMATHEMATICALSTUDYOFPROBABILITYANDSTATISTICSHAVEACOMMONHISTORICALORIGININATTEMPTSTOUNDERSTANDANDPRESUMABLYTOWINGAMESOFCHANCETHESTUDYOFSIMPLECOINIPPINGGAMES,FOREXAMPLE,HASBEENUNUSUALLYPRODUCTIVEINMATHEMATICSANDINILLUMINATINGCERTAINCHARACTERISTICSOFHUMANBEHAVIORTHATTHEGAMESEXHIBITTWOSTATISTICALCONCEPTSTHATORIGINATEDINSUCHGAMES,ANDWILLBEQUITEUSEFULINTHEREMAINDEROFTHISCHAPTER,AREPROBABILITYANDEXPECTEDVALUETHEPROBABILITYOFAREPETITIVE1EVENTHAPPENINGIS,ROUGHLYSPEAKING,THERELATIVEFREQUENCYWITHWHICHITWILLOCCURFOREXAMPLE,TOSAYTHATTHEPROBABILITYOFOBTAININGAHEADONTHEIPOFAFAIRCOINISONEHALFMEANSTHATONEWOULDEXPECTTHATIFAFAIRCOINWEREIPPEDALARGENUMBEROFTIMES,AHEADWOULDAPPEARINAPPROXIMATELYONEHALFOFTHETRIALSSIMILARLY,THEPROBABILITYOFROLLINGA2ONASINGLEDIEISONESIXTHINAPPROXIMATELYONEOUTOFEVERYSIXROLLS,A2WILLCOMEUPSUPPOSETHATALOTTERYOFFERSNDISTINCTPRIZESSOMEOFWHICHMAYBE0OREVENNEGATIVE,X1,X2,XN,ANDTHATTHEPROBABILITIESOFWINNINGTHESEPRIZESAREH92661,H92662,H9266NIFWEASSUMETHATONEANDONLYONEPRIZEWILLBEAWARDEDTOAPLAYER,ITMUSTBETHECASETHATH20913NIH110051H9266IH11005181EQUATION81SIMPLYSAYSTHATOURLISTINDICATESALLPOSSIBLEOUTCOMESOFTHELOTTERYANDTHATONEOFTHESEHASTOOCCURTOPROVIDEAMEASUREOFTHEAVERAGEPAYOFFINTHISLOTTERY,WEDENEEXPECTEDVALUEASFOLLOWSEXPECTEDVALUEFORALOTTERYXWITHPRIZESX1,X2,XNANDPROBABILITIESOFWINNINGH92661,H92662,H9266N,THEEXPECTEDVALUEOFTHELOTTERYIS2EXPECTEDVALUEH11005EXH11005H92661X1H11001H92662X2H11001H11001H9266NXN82H11005H20913NIH110051H9266IXIDEFINITION198PARTIIICHOICEUNDERUNCERTAINTY1FORREPETITIVEEVENTS,PROBABILITYISANOBJECTIVELYDENEDCONCEPTINDIVIDUALSMAYALSOATTACHSUBJECTIVEPROBABILITIESTONONRECURRINGEVENTSFORTHEMOSTPART,WEDONOTDISTINGUISHBETWEENTHESETWOTYPESOFPROBABILITYESTIMATES2IFTHESITUATIONBEINGEXAMINEDHASCONTINUOUSOUTCOMESFOREXAMPLE,THECHANGEINTHEPRICEOFASTOCKMEASUREDVERYPRECISELY,THENWENEEDTOMODIFYTHISDENITIONABITIFTHEPROBABILITYTHATANOUTCOMEOFSUCHARANDOMEVENTXISINASMALLINTERVALDXISGIVENBYFXDX,THENEQUATION81CANBEMODIEDASH20885H11009H11002H11009FXDXH110051INTHISCASETHEEXPECTEDVALUEOFXISGIVENBYEXH11005H20885H11009H11002H11009XFXDXINMANYSITUATIONSFOREXAMPLE,WHENXHASANORMALDISTRIBUTION,MANIPULATIONOFSUCHEXPECTEDVALUESCANBEMUCHSIMPLERTHANFORTHEDISCRETECASEREPRESENTEDINEQUATION82SEETHEEXTENSIONSTOTHISCHAPTERFORSOMEILLUSTRATIONSTHEEXPECTEDVALUEOFTHELOTTERYISAWEIGHTEDSUMOFTHEPRIZES,WHERETHEWEIGHTSARETHERESPECTIVEPROBABILITIESITISSIMPLYTHESIZEOFTHEPRIZETHATTHEPLAYERWILLWINONTHEAVERAGEFOREXAMPLE,SUPPOSETHATJONESANDSMITHAGREETOIPACOINONCEIFAHEADCOMESUP,JONESWILLPAYSMITH1IFATAIL,SMITHWILLPAYJONES1FROMSMITHSPOINTOFVIEW,THEREARETWOPRIZESINTHISGAMEFORAHEAD,X1H11005H110011FORATAIL,X2H11005H110021,WHERETHEMINUSSIGNINDICATESTHATSMITHMUSTPAYFROMJONESSPOINTOFVIEW,THEGAMEISEXACTLYTHESAMEEXCEPTTHESIGNSOFTHEOUTCOMESAREREVERSEDTHUS,THEEXPECTEDVALUEOFTHEGAMEISX1H11001X2H110051H11001H110021H11005083THEGAMEHASANEXPECTEDVALUEOF0IFTHEGAMEWERETOBEPLAYEDALARGENUMBEROFTIMES,ITISNOTLIKELYTHATEITHERPLAYERWOULDCOMEOUTVERYFARAHEADNOWSUPPOSETHATTHEPRIZESOFTHEGAMEWERECHANGEDSOTHATAGAINFROMSMITHSPOINTOFVIEWX1H1100510,X2H11005H110021SMITHWILLWIN10IFAHEADCOMESUPBUTWILLLOSEONLY1IFATAILAPPEARSTHEEXPECTEDVALUEOFTHISGAMEISX1H11001X2H1100510H11001H110021H110055H1100250H1100545084IFTHISGAMEISPLAYEDMANYTIMES,SMITHWILLCERTAINLYENDUPTHEBIGWINNERINFACT,SMITHMIGHTBEWILLINGTOPAYJONESSOMETHINGFORTHEPRIVILEGEOFPLAYINGTHEGAMEGAMESSUCHASTHOSEINEQUATION83,WHICHHAVEANEXPECTEDVALUEOF0,ORTHOSEINEQUATION84,WHICHCOSTTHEIREXPECTEDVALUESHERE,PRECISELY450FORTHERIGHTTOPLAY,ARECALLEDACTUARIALLYFAIRGAMESACOMMONOBSERVATIONISTHAT,INMANYSITUATIONS,PEOPLEREFUSETOPARTICIPATEINACTUARIALLYFAIRGAMESTHISPOINTISCENTRALTOUNDERSTANDINGDEVELOPMENTSINTHETHEORYOFUNCERTAINTYANDISTAKENUPINTHENEXTSECTIONFAIRGAMESANDTHEEXPECTEDUTILITYHYPOTHESISPEOPLEAREGENERALLYUNWILLINGTOPLAYFAIRGAMES3IMAYATTIMESAGREETOIPACOINFORSMALLAMOUNTSOFMONEY,BUTIFIWEREOFFEREDTHECHANCETOWAGER1000ONONECOINIP,IWOULDUNDOUBTEDLYREFUSESIMILARLY,PEOPLEMAYSOMETIMESPAYASMALLAMOUNTOFMONEYTOPLAYANACTUARIALLYUNFAIRGAMESUCHASASTATELOTTERY,BUTTHEYWILLAVOIDPAYINGAGREATDEALTOPLAYRISKY,BUTFAIRGAMESSTPETERSBURGPARADOXACONVINCINGEXAMPLEISTHE“STPETERSBURGPARADOX,”WHICHWASRSTINVESTIGATEDRIGOROUSLYBYTHEMATHEMATICIANDANIELBERNOULLIINTHEEIGHTEENTHCENTURY4INTHE1H500721H500721H500721H500721H500721H500721H500721H50072CHAPTER8EXPECTEDUTILITYANDRISKAVERSION1993THEGAMESDISCUSSEDHEREAREASSUMEDTOYIELDNOUTILITYINTHEIRPLAYOTHERTHANTHEPRIZESHENCE,THEOBSERVATIONTHATMANYINDIVIDUALSGAMBLEAT“UNFAIR”ODDSISNOTNECESSARILYAREFUTATIONOFTHISSTATEMENTRATHER,SUCHINDIVIDUALSCANREASONABLYBEASSUMEDTOBEDERIVINGSOMEUTILITYFROMTHECIRCUMSTANCESASSOCIATEDWITHTHEPLAYOFTHEGAMEITISTHEREFOREPOSSIBLETODIFFERENTIATETHECONSUMPTIONASPECTOFGAMBLINGFROMTHEPURERISKASPECT4THEORIGINALBERNOULLIPAPERHASBEENREPRINTEDASDBERNOULLI,“EXPOSITIONOFANEWTHEORYONTHEMEASUREMENTOFRISK,”ECONOMETRICA22JANUARY19542336STPETERSBURGPARADOXTHEFOLLOWINGGAMEISPROPOSEDACOINISIPPEDUNTILAHEADAPPEARSIFAHEADRSTAPPEARSONTHENTHIP,THEPLAYERISPAID2NTHISGAMEHASANINNITENUMBEROFOUTCOMESACOINMIGHTBEIPPEDFROMNOWUNTILDOOMSDAYANDNEVERCOMEUPAHEAD,ALTHOUGHTHELIKELIHOODOFTHISISSMALL,BUTTHERSTFEWCANEASILYBEWRITTENDOWNIFXIREPRESENTSTHEPRIZEAWARDEDWHENTHERSTHEADAPPEARSONTHEITHTRIAL,THENX1H110052,X2H110054,X3H110058,XNH110052N85THEPROBABILITYOFGETTINGAHEADFORTHERSTTIMEONTHEITHTRIALISH20898H500712H5007H20899IITISTHEPROBABILITYOFGETTINGIH110021TAILSANDTHENAHEADHENCETHEPROBABILITIESOFTHEPRIZESGIVENINEQUATION95AREH92661H11005,H92662H11005,H92663H11005,H9266NH1100586THEEXPECTEDVALUEOFTHESTPETERSBURGPARADOXGAMEISINNITEEXPECTEDVALUEH11005H20913H11009IH110051H9266IXIH11005H20913H11009IH1100512I87H110051H110011H110011H11001H110011H11001H11005H11009SOMEINTROSPECTION,HOWEVER,SHOULDCONVINCEANYONETHATNOPLAYERWOULDPAYVERYMUCHMUCHLESSTHANINFINITYTOPLAYTHISGAMEIFICHARGED1BILLIONTOPLAYTHEGAME,IWOULDSURELYHAVENOTAKERS,DESPITETHEFACTTHAT1BILLIONISSTILLCONSIDERABLYLESSTHANTHEEXPECTEDVALUEOFTHEGAMETHIS,THEN,ISTHEPARADOXBERNOULLISGAMEISINSOMESENSENOTWORTHITSINFINITEEXPECTEDDOLLARVALUEEXPECTEDUTILITYBERNOULLISSOLUTIONTOTHISPARADOXWASTOARGUETHATINDIVIDUALSDONOTCAREDIRECTLYABOUTTHEDOLLARPRIZESOFAGAMERATHERTHEYRESPONDTOTHEUTILITYTHESEDOLLARSPROVIDEIFWEASSUMETHATTHEMARGINALUTILITYOFINCOMEDECLINESASINCOMEINCREASES,THESTPETERSBURGGAMEMAYCONVERGETOANITEEXPECTEDUTILITYVALUETHATPLAYERSWOULDBEWILLINGTOPAYFORTHERIGHTTOPLAYBERNOULLITERMEDTHISEXPECTEDUTILITYVALUETHEMORALVALUEOFTHEGAMEBECAUSEITREPRESENTSHOWMUCHTHEGAMEISWORTHTOTHEINDIVIDUALBECAUSEUTILITYMAYRISELESSRAPIDLYTHANTHEDOLLARVALUEOFTHEPRIZES,ITISPOSSIBLETHATAGAMESMORALVALUEWILLFALLSHORTOFITSMONETARYEXPECTEDVALUEEXAMPLE81BERNOULLISSOLUTIONTOTHEPARADOXSUPPOSE,ASDIDBERNOULLI,THATTHEUTILITYOFEACHPRIZEINTHESTPETERSBURGPARADOXISGIVENBYUXIH11005LNXI881H50072I1H50072N1H500781H500741H50072200PARTIIICHOICEUNDERUNCERTAINTYTHISNATURALLOGARITHMICUTILITYFUNCTIONEXHIBITSDIMINISHINGMARGINALUTILITYTHATIS,U1H110220,BUTUH11033H110210,ANDTHEEXPECTEDUTILITYVALUEOFTHISGAMECONVERGESTOANITENUMBEREXPECTEDUTILITYH11005H20913H11009IH110051H9266IUXI89H11005H20913H11009IH110051LN2ISOMEMANIPULATIONOFTHISEXPRESSIONYIELDS5THERESULTTHATTHEEXPECTEDUTILITYVALUEOFTHISGAMEIS139ANINDIVIDUALWITHTHISTYPEOFUTILITYFUNCTIONMIGHTTHEREFOREBEWILLINGTOINVESTRESOURCESTHATOTHERWISEYIELDUPTO139UNITSOFUTILITYACERTAINWEALTHOFABOUT4PROVIDESTHISUTILITYINPURCHASINGTHERIGHTTOPLAYTHISGAMEASSUMINGTHATTHEVERYLARGEPRIZESPROMISEDBYTHESTPETERSBURGPARADOXENCOUNTERDIMINISHINGMARGINALUTILITYTHEREFOREPERMITTEDBERNOULLITOOFFERASOLUTIONTOTHEPARADOXQUERYDOESBERNOULLISSOLUTIONREALLY“SOLVE”THEPARADOXHOWWOULDYOUREDENETHEPRIZESINTHISGAMESOTHATTHEGAMEWOULDHAVEANINNITEEXPECTEDUTILITYVALUEUSINGTHELOGARITHMICUTILITYFUNCTIONTHEVONNEUMANNMORGENSTERNTHEOREMINTHEIRBOOKTHETHEORYOFGAMESANDECONOMICBEHAVIOR,JOHNVONNEUMANNANDOSCARMORGENSTERNDEVELOPEDMATHEMATICALMODELSFOREXAMININGTHEECONOMICBEHAVIOROFINDIVIDUALSUNDERCONDITIONSOFUNCERTAINTY6TOUNDERSTANDTHESEINTERACTIONS,ITWASNECESSARYRSTTOINVESTIGATETHEMOTIVESOFTHEPARTICIPANTSINSUCH“GAMES”BECAUSETHEHYPOTHESISTHATINDIVIDUALSMAKECHOICESINUNCERTAINSITUATIONSBASEDONEXPECTEDUTILITYSEEMEDINTUITIVELYREASONABLE,THEAUTHORSSETOUTTOSHOWTHATTHISHYPOTHESISCOULDBEDERIVEDFROMMOREBASICAXIOMSOF“RATIONALE”BEHAVIORTHEAXIOMSREPRESENTANATTEMPTBYTHEAUTHORSTOGENERALIZETHEFOUNDATIONSOFTHETHEORYOFINDIVIDUALCHOICETOCOVERUNCERTAINSITUATIONSALTHOUGHMOSTOFTHESEAXIOMSSEEMEMINENTLYREASONABLEATRSTGLANCE,MANYIMPORTANTQUESTIONSABOUTTHEIRTENABILITYHAVEBEENRAISEDWEWILLNOTPURSUETHESEQUESTIONSHERE,HOWEVER7THEVONNEUMANNMORGENSTERNUTILITYINDEXTOBEGIN,SUPPOSETHATTHEREARENPOSSIBLEPRIZESTHATANINDIVIDUALMIGHTWINBYPARTICIPATINGINALOTTERYLETTHESEPRIZESBEDENOTEDBYX1,X2,XNANDASSUME1H50072ICHAPTER8EXPECTEDUTILITYANDRISKAVERSION2015PROOFEXPECTEDUTILITYH11005H20913H11009IH110051H110801/2IH11080LN2H11005LN2H20913H11009IH110051I/2IBUTTHEVALUEOFTHISNALINNITESERIESCANBESHOWNTOBE20HENCE,EXPECTEDUTILITYH110052LN2H110051396JVONNEUMANNANDOMORGENSTERN,THETHEORYOFGAMESANDECONOMICBEHAVIORPRINCETON,NJPRINCETONUNIVERSITYPRESS,1944THEAXIOMSOFRATIONALITYINUNCERTAINSITUATIONSAREDISCUSSEDINTHEAPPENDIX7FORADISCUSSIONOFSOMEOFTHEISSUESRAISEDINTHEDEBATEOVERTHEVONNEUMANNMORGENSTERNAXIOMS,SEEMARKJMACHINA,“CHOICEUNDERUNCERTAINTYPROBLEMSSOLVEDANDUNSOLVED,”JOURNALOFECONOMICPERSPECTIVESSUMMER1987121154THATTHESEHAVEBEENARRANGEDINORDEROFASCENDINGDESIRABILITYX1ISTHEREFORETHELEASTPREFERREDPRIZEFORTHEINDIVIDUAL,ANDXNISTHEMOSTPREFERREDPRIZENOWASSIGNARBITRARYUTILITYNUMBERSTOTHESETWOEXTREMEPRIZESFOREXAMPLE,ITMIGHTBECONVENIENTTOASSIGNUX1H110050UXNH110051,810BUTANYOTHERPAIROFNUMBERSWOULDDOEQUALLYWELL8USINGTHESETWOVALU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YCHOOSETHEONEWITHASMALLERVARIABILITYOFRETURNINTUITIVELY,THEREASONBEHINDTHISISTHATWEUSUALLYASSUMETHATTHEMARGINALUTILITYFROMEXTRADOLLARSOFPRIZEMONEYTHATIS,WEALTHDECLINESASTHEPRIZESGETLARGERAIPOFACOINFOR1000PROMISESARELATIVELYSMALLGAINOFUTILITYIFYOUWINBUTALARGELOSSOFUTILITYIFYOULOSEABETOFONLY1IS“INCONSEQUENTIAL,”ANDTHEGAININUTILITYFROMAWINAPPROXIMATELYCOUNTERBALANCESTHEDECLINEINUTILITYFROMALOSS10RISKAVERSIONANDFAIRBETSTHISARGUMENTISILLUSTRATEDINFIGURE81HEREWREPRESENTSANINDIVIDUALSCURRENTWEALTHANDUWISAVONNEUMANNMORGENSTERNUTILITYINDEXTHATREECTSHOWHEORSHEFEELSABOUTVARIOUSLEVELSOFWEALTHUWISDRAWNASACONCAVEFUNCTIONOFWTOREECTTHEASSUMPTIONOFADIMINISHINGMARGINALUTILITYITISASSUMEDTHATOBTAININGANEXTRADOLLARADDSLESSTOENJOYMENTASTOTALWEALTHINCREASESNOWSUPPOSETHISPERSONISOFFEREDTWOFAIRGAMBLESA5050CHANCEOFWINNINGORLOSINGHORA5050CHANCEOFWINNINGORLOSING2HTHEUTILITYOFPRESENTWEALTHISUWTHEEXPECTEDUTILITYIFHEORSHEPARTICIPATESINGAMBLE1ISGIVENBYUHWUHWH11005UWH11001HH11001UWH11002H,817ANDTHEEXPECTEDUTILITYOFGAMBLE2ISGIVENBYU2HWU2HWH11005UWH110012HH11001UWH110022H818ITISGEOMETRICALLYCLEARFROMTHEGURETHAT11UWH11022UHWH11022U2HW819THISPERSONTHEREFOREWILLPREFERCURRENTWEALTHTOTHATWEALTHCOMBINEDWITHAFAIRGAMBLEANDWILLPREFERASMALLGAMBLETOALARGEONETHEREASONFORTHISISTHATWINNINGAFAIRBETADDSTOENJOYMENTLESSTHANLOSINGHURTSALTHOUGHTHEPRIZESAREEQUALINEXPECTEDVALUETERMS,INUTILITYTERMSTHELOSSISMORESERIOUS1H500721H500721H500721H50072204PARTIIICHOICEUNDERUNCERTAINTY9OFTENTHESTATISTICALCONCEPTOF“VARIANCE”ISUSEDASAPROXYFORRISKALTHOUGH,WEWILLNOTDISCUSSTHISSTATISTICALNOTIONINTHEBODYOFTHISCHAPTER,ITISDENEDINTHECHAPTERSEXTENSIONSANDUSEDINAFEWPROBLEMS10ANALTERNATIVE,MOREGENERALDENITIONOFRISKAVERSIONISTHATEUWH11021UEWFORANYRANDOMLYDISTRIBUTEDWEALTH,WDIMINISHINGMARGINALUTILITYENSURESTHISCONDITION11TOSEEWHYTHEEXPECTEDUTILITIESFORBETHANDBET2HARETHOSESHOWN,NOTICETHATTHESEEXPECTEDUTILITIESARESIMPLYTHEAVERAGEOFTHEUTILITIESFROMAFAVORABLEANDANUNFAVORABLEOUTCOMEBECAUSEWISHALFWAYBETWEENWH11001HANDWH11002H,UHISALSOHALFWAYBETWEENUWH11001HANDUWH11002HRISKAVERSIONANDINSURANCEASAMATTEROFFACT,THISPERSONMIGHTBEWILLINGTOPAYSOMEAMOUNTTOAVOIDPARTICIPATINGINANYGAMBLEATALLNOTICETHATACERTAINWEALTHOFWH33526PROVIDESTHESAMEUTILITYASDOESPARTICIPATINGINGAMBLE1THEINDIVIDUALWILLBEWILLINGTOPAYANYTHINGUPTOWH11002WH33526TOAVOIDPARTICIPATINGINTHEGAMBLETHISUNDOUBTEDLYEXPLAINSWHYPEOPLEBUYINSURANCETHEYAREGIVINGUPASMALL,CERTAINAMOUNTTHEPREMIUMTOAVOIDTHERISKYOUTCOMETHEYAREBEINGINSUREDAGAINSTTHEPREMIUMAPERSONPAYSFORAUTOMOBILECOLLISIONINSURANCE,FOREXAMPLE,PROVIDESAPOLICYTHATAGREESTOREPAIRHISORHERCARSHOULDANACCIDENTOCCURTHEWIDESPREADUSEOFSUCHINSURANCEWOULDSEEMTOIMPLYTHATAVERSIONTORISKISQUITECOMMONHENCE,WEINTRODUCETHEFOLLOWINGDENITIONRISKAVERSIONANINDIVIDUALWHOALWAYSREFUSESFAIRBETSISSAIDTOBERISKAVERSEIFINDIVIDUALSEXHIBITADIMINISHINGMARGINALUTILITYOFWEALTH,THEYWILLBERISKAVERSEASACONSEQUENCE,THEYWILLBEWILLINGTOPAYSOMETHINGTOAVOIDTAKINGFAIRBETSDEFINITIONCHAPTER8EXPECTEDUTILITYANDRIS