Lecture 5 2013 Choice under Uncertainty · 2016. 12. 3. · LECTURE 5 MICROECONOMIC THEROY CONSUMER...
Transcript of Lecture 5 2013 Choice under Uncertainty · 2016. 12. 3. · LECTURE 5 MICROECONOMIC THEROY CONSUMER...
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LECTURE 5MICROECONOMIC THEROY
CONSUMER THEORYChoice under Uncertainty
(MWG chapter 6, sections A-C, and Cowell chapter 8)
Lecturer: Andreas Papandreou
1
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Introductionp Contents
n Expectedutilitytheoryn Measuresofriskaversionn Measuresofrisk
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Introductionp Until now, we have been concerned with choices
under certainty.n If I choose A, the outcome is with certainty CA and my
utility is with certainty u(CA).n If I choose B, the outcome is with certainty CB and my
utility is with certainty u(CB).n A B u(CA) ≥ u(CB)
p What if A and B are not certainties, but distributions over outcomes?
! Û
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IntroductionEXAMPLEp Supposeweareconsideringtwodifferentuncertainalternatives,eachofwhichoffersadifferentdistributionoverthreeoutcomes:n IbuyyouatriptoBermudan youpayme$500n youdoallmyundergraduatetutorialsofmicro
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Introductionp TheprobabilityofeachoutcomeunderalternativesAandB
arethefollowing:
p Wewouldliketoexpressyourutilityforthesetwoalternativesintermsoftheutilityyouassigntoeachindividualoutcomeandtheprobabilitythattheyoccur
Bermuda -$500 Do micro tutorials
A 0.3 0.4 0.3B 0.2 0.7 0.1
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Introductionp Supposeyouassign:
n ValueuB tothetriptoBermudan Valueum topayingmethemoneyn Valueuttodoingthetutorials
p Andweknow:n ProbabilitypB tothetriptoBermudan ProbabilitypB topayingmethemoneyn Probabilitypt todoingthetutorials
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ExpectedUtilityp Itwouldbeveryniceifwecouldexpressyourutilityforeachalternativebymultiplyingeachofthesenumbersbytheprobabilityoftheoutcomeoccurring,andsummingup.Thatis:
U (A)=0.3uB +0.4um+0.3utU(B)=0.2uB +0.7um +0.1ut.
Or,ingeneral,theexpectedutilityofanalternativewouldbe
EU (A)=pBuB +pmum+ptut
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ExpectedUtilityPrinciplep Moregenerally:
C1 " u(C1)d1 " u(d1)AC2 " u(C2)Bd2 " u(d2)
..Cn " u(Cn) dm " u(dm)
p1+p2+…+pn=1q1+q2+…+qm=1DoIchooseAorB?ExpectedUtilityPrinciple: ABEU(A)≥EU(B)EU (A)=p1u(C1)+p2u(C2)+…+pnu(Cn)EU (B)=q1u(d1)+q2u(d2)+…+qnu(dm)
! Û
p1
p2
pn
q1
q2
qm
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ExpectedUtilityPrinciplep Theonlydifferenceisthatwemaximizeexpectedutility.p Inch.3ofMWGwebasedouranalysisontheassumption
thataconsumerhasrationalpreferencesp However,theassumptionofrationalpreferencesover
uncertainoutcomesisnotsufficienttorepresentthesepreferencesbyautilityfunctionthathastheexpectedutilityform
p Tobeabletodoso,wehavetoplaceadditionalstructureonpreferences
p Thenweshowhowutilityfunctionsoftheexpectedutilityformcanbeusedtostudybehaviorunderuncertainty,anddrawtestableimplications
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Assumptionsonpreferences:1.Rationality
1.Theindividualhascompleteandtransitivepreferencesoverdifferentoutcomes(rationality)n ForanyCi,Cj Ci Cj orCj Ci(orboth)
Þ !!
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Assumptionsonpreferences2.Reductionofcompoundlotteries
2.Reductionofcompoundlotteries(orconsequentialistpreferences)
Alotteryisaprobabilitydistributionoverasetofpossibleoutcomes.Asimplelottery isavectorL=(p1,p2,…,pN)suchthatpn≥0foralln andΣnpn=1.
Inacompoundlottery anoutcomemayitselfbeasimplelottery
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AssumptionsonpreferencesReductionofcompoundlotteries
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AssumptionsonpreferencesReductionofcompoundlotteries
p Reductionofcompoundlotteries(consequentialistpreferences):Consumerscareonlyaboutthedistributionoverfinaloutcomes,notwhetherthisdistributioncomesaboutasaresultofasimplelottery,oracompoundlottery.Inotherwords,theconsumerisindifferentbetweenanytwocompoundlotteriesthatcanbereducedtothesamesimplelottery.Thispropertyisoftencalledreductionofcompoundlotteries.
p Becauseofthereductionproperty,wecanconfineourattentiontothesetofallsimplelotteries.
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AssumptionsonpreferencesReductionofcompoundlotteries
C1 C1
C2 C2
A ~ BC3 C3
p Anotherwaytothinkaboutthereductionpropertyisthatwe’reassumingthereisnoprocess-orientedutility.Consumersdonotenjoytheprocessofthegamble,onlytheoutcome,eliminatingthe“funofthegamble”insettingslikecasinos.
p Onlytheoutcomematters,nottheprocess.
p1p1·p2
p4
p3
p2
p1·p3
p4
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Assumptionsonpreferences3.Theindependenceaxiom
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AssumptionsonpreferencesTheindependenceaxiom
p SupposethatIofferyouthechoicebetweenthefollowingtwoalternatives:
L:$5withprobability1/5,0withprobability4/5L’:$12withprobability1/10,0withprobability9/10
p SupposeyoupreferLtoL’.Nowconsiderthefollowingalternative.Iflipacoin.Ifitcomesupheads,IofferyouthechoicebetweenLandL’.Ifitcomesuptails,yougetnothing.WhattheindependenceaxiomsaysisthatifIaskyoutochooseeitherLorL’beforeIflipthecoin,yourpreferenceshouldbethesameasitwaswhenIdidn’tflipthecoin.
p Independenceofirrelevantalternatives
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Assumptionsonpreferences4.Continuity
SupposeC1 isthebestoutcomeSupposeCn istheworstoutcome
ForanyoutcomeCi betweenbestandworst,therewillbesomeprobabilitypi,suchthatweareindifferentbetween:
C1 $1000Ci ~
Cn $0$1000
e.g.$400~$0
100%pi
1-pipi
1-pi
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Assumptionsonpreferencesp Theaboveassumptionscombinedmeanthatanylotterymaybewrittenasasimplerlotterythatonlyinvolvesthebestandtheworstoutcome.
p Example:
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Assumptionsonpreferences5.Monotonicity
C1(best) C1 (best)
A BCm(worst) Cm (worst)
ABiffp ≥q
p q
1-p 1-q
!
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ExpectedUtilityTheoremp Theexpectedutilitytheoremsaysthatifaconsumer’s
preferencesoversimplelotteriesarerational,continuous,andexhibitthereductionandindependenceproperties,thenthereisautilityfunctionoftheexpectedutilityformthatrepresentsthosepreferences.
Thatis,therearenumbersu1,...,uN suchthat
å=n
nnupLEU )(
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ExpectedUtilityTheorem:examplep Takeumbrella/Nottakeumbrellap Preferencesoftheindividual:
u(noumbrella,sunny)=10=u1u(umbrella,rains)=8=u2u(umbrella,sunny)=6=u3u(noumbrella,rains)=0=u4
1) Ifyouknowit’sgoingtorain:u2 >u4" UMB2) Ifyouknowit’snotgoingtorain:u1 >u3" noUMB
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ExpectedUtilityTheorem:examplep Takeumbrella/Nottakeumbrellap Preferencesoftheindividual:
u(noumbrella,sunny)=10=u1u(umbrella,rains)=8=u2u(umbrella,sunny)=6=u3u(noumbrella,rains)=0=u4
3)UtilitymaximizationifPrain =0.6.EU(UMB)=0.6*8+0.4*6=7.2EU(NOUMB)=0.6*0+.4*10=4
ItakeanumbrellabecauseEU(UMB)>EU(NOUMB)
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ExpectedUtilityTheorem:examplep AmIallowedtouseordinalutilityfunctionswhenIworkwithuncertainty?
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ExpectedUtilityTheorem:examplep AmIallowedtouseordinalutilityfunctionswhenIworkwithuncertainty?
NO
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ExpectedUtilityTheorem:examplep Takeumbrella/Nottakeumbrellap Preferencesoftheindividual:
u(noumbrella,sunny)=100=u1u(umbrella,rains)=8 =u2u(umbrella,sunny)=6 =u3u(noumbrella,rains)=0 =u4
Thisisstillautilityfunctionpreservingtheorderofpreferences(ordinal)
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ExpectedUtilityTheorem:examplep Takeumbrella/Nottakeumbrellap Preferencesoftheindividual:
u(noumbrella,sunny)=100=u1u(umbrella,rains)=8 =u2u(umbrella,sunny)=6 =u3u(noumbrella,rains)=0 =u4
3)UtilitymaximizationifPrain =0.6.EU(UMB)=0.6*8+0.4*6=7.2EU(NOUMB)=0.6*0+.4*100 =40
Don’ttakeanumbrellabecauseEU(UMB)< EU(NOUMB)!!!
Thisisstillautilityfunctionpreservingtheorderofpreferences(ordinal)
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Von-Neumann-MorgensternUF(vN-M)p TheExpectedUtilityFormispreservedonlybypositivelinear
transformations.IfU (.)andV (.)areutilityfunctionsrepresenting,andU ()hastheexpectedutilityform,thenV (.)alsohastheexpectedutilityformifandonlyiftherearenumbersa >0andb suchthat:
U (L)=aV (L)+b.Inotherwords,theexpectedutilitypropertyispreservedbypositivelineartransformations,butanyothertransformationofU(.)doesnotpreservethisproperty.
p Wewillcalltheutilityfunctionoftheexpectedutilityformavon-Neumann-Morgenstern(vNM)utilityfunction.
!
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Von-Neumann-MorgensternUF(vN-M)p Inthepreviousexample
u(noumbrella,sunny)=10=u1u(umbrella,rains)=8 =u2u(umbrella,sunny)=6 =u3u(noumbrella,rains)=0 =u4
Couldbetransformedtou(noumbrella,sunny)=100=v1u(umbrella,rains)=80 =v2u(umbrella,sunny)=60 =v3u(noumbrella,rains)=0 =v4
Wherev(.)=α*u(.)+βwithα=10 andβ=0
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Von-Neumann-MorgensternUF(vN-M)p Inthepreviousexample
u(noumbrella,sunny)=10=u1u(umbrella,rains)=8 =u2u(umbrella,sunny)=6 =u3u(noumbrella,rains)=0 =u4
Alternatively,Icouldsettoputilityv1 =100,worstutilityv4 =5andcalculatewhattheothertwoutilitiesshouldbesothatIhavealineartransformation,100=a*10+β and5=α*0+β,sothatα=9.5andβ=5.u(noumbrella,sunny)=100=v1u(umbrella,rains)=9.5*8+5 =v2u(umbrella,sunny)=9.5*6+5 =v3u(noumbrella,rains)=5 =v4
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Von-Neumann-MorgensternUF(vN-M)p ThenumbersassignedbythevN-Mutilityfunctionhavecardinalsignificance.
p Supposeu(A)=30u(B)=20u(C)=10
IsAthreetimesbetterthanC?Answer:NO(a>0)
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Von-Neumann-MorgensternUF(vN-M)p ThevN-Mutilityfunctioniscardinalinthesensethatutilitydifferences arepreserved.
p Forexamplestartwithu(A)=30,u(B)=20,u(C)=10
p Applylineartransformationwithα=2andβ=1,v(A)=61,v(B)=41,v(C)=21
p AispreferredtoBasmuchasBispreferredtoC.
10 10
2020
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Von-Neumann-MorgensternUF(vN-M)
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Von-Neumann-MorgensternUF(vN-M)
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Von-Neumann-MorgensternUF(vN-M)-example
Exercise:assumeanindividualwithpreferencesA B C D.ThisindividualisindifferentbetweenB andthelottery(A,D;0.4,0.6).AlsosheisindifferentbetweenC andthelottery(B, D;0.2,0.8).ConstructasetofvN-Mutilitynumbersforthefoursituations.
!! !
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Von-Neumann-MorgensternUF(vN-M)-example
Exercise:assumeanindividualwithpreferencesABCD.ThisindividualisindifferentbetweenBandthelottery(A,D;0.4,0.6).AlsosheisindifferentbetweenCandthelottery(B,D;0.2,0.8).ConstructasetofvN-Mutilitynumbersforthefoursituations.
AnswerBydefinitionoftheutilityfunctionUA>UB>UC>UD.AB̴D,thereforeUB=0.4*UA+0.6*UD
BC̴D,thereforeUC=0.2*UB+0.8*UD
!! !
0.4
0.6
0.2
0.8
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Von-Neumann-MorgensternUF(vN-M)-Example
p Choosejusttwoutilitylevels(α u(.)+ β):givesustwodegreesoffreedom.SoUA =1andUD=0,thensolveforUB andUC
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Objectionswiththetheoryofexpectedutility:Allaisparadox
p Allais(1953):presentparticipantswithtwodifferentexperiments.
p Firstexperiment:choosebetweenAandB
$2,500A$2,400B$2,400
$0
WouldyouchooseAorB?
33%
66%
1%
100%
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Objectionswiththetheoryofexpectedutility:Allaisparadox
p Allais(1953):presentparticipantswithtwodifferentexperiments.
p Firstexperiment:choosebetweenAandB
$2,500A$2,400B$2,400
$0
MostparticipantschooseB,thereforetheymustconsiderU(2,400)>0.33U(2,500)+0.66(2,400)+0.01U(0)
33%
66%
1%
100%
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Objectionswiththetheoryofexpectedutility:Allaisparadox
p Allais(1953):presentparticipantswithtwodifferentexperiments.
p Secondexperiment:choosebetweenCandD
$2,500$2,400CD
$0$0
MostparticipantschooseC,thereforetheymustconsider0.33*U(2,500)+0.67*U(0)>0.34U(2,400)+0.66U(0)
33%
67%
34%
66%
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Objectionswiththetheoryofexpectedutility:Allaisparadox
p ThesechoicesdonotaccordwiththeExpectedutilitytheory.
U(2,400)>0.33U(2,500)+0.66U(2,400)+0.01U(0)(1)0.33*U(2,500)+0.67*U(0)>0.34U(2,400)+0.66U(0)(2)
Thesetwosayexactlytheoppositething!
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Objectionswiththetheoryofexpectedutility:Allaisparadox
p Possible explanationsn U(0 / when $2,500 is also available) ≠
U(0 / if the top price is $10) because of the regret you would feel.
n People are not rationaln People cannot process very small/high
probabilitiesn “framing effect”: equivalent descriptions of a
decision problem lead to systematically different decisions
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Objectionswiththetheoryofexpectedutility:the“framingeffect”
MoreontheframingeffectObjectsdescribedintermsofapositivelyvalencedproportionaregenerallyevaluatedmorefavorablythanobjectsdescribedintermsofthecorrespondingnegativelyvalencedproportion.Forexample,inonestudy,beefdescribedas“75%lean”wasgivenhigherratingsthanbeefdescribedas“25%fat”(LevinandGaeth1988)
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Objectionswiththetheoryofexpectedutility:the“framingeffect”
p Thebest-knownriskychoiceframingproblemistheso-called“AsianDiseaseProblem”(TverskyandKahneman1981).Init,subjectsfirstreadthefollowingbackgroundblurb:n ImaginethattheU.S.ispreparingfortheoutbreakofanunusual
Asiandisease,whichisexpectedtokill600people.Onepossibleprogramtocombatthediseasehasbeenproposed.Assumethattheexactscientificestimateoftheconsequencesofthisprogramisasfollows:
SomesubjectsarethenpresentedwithoptionsAandB:n A:Ifthisprogramisadopted,200peoplewillbesaved.n B:Ifthisprogramisadopted,thereisaone-thirdprobabilitythat
600peoplewillbesavedandatwo-thirdsprobabilitythatnopeoplewillbesaved.
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Objectionswiththetheoryofexpectedutility:the“framingeffect”
p OthersubjectsarepresentedwithoptionsCandD:n C:Ifthisprogramisadopted,400peoplewilldie.n D:Ifthisprogramisadopted,thereisaone-thirdprobabilitythatnobodywilldieandatwo-thirdsprobabilitythat600peoplewilldie.
TherobustexperimentalfindingisthatsubjectstendtopreferthesurethingwhengivenoptionsAandB,buttendtopreferthegamblewhengivenoptionsCandD.Note,however,thatoptionsAandCareequivalent,asareoptionsBandD.Subjectsthusappeartoberisk-averseforgainsandrisk-seekingforlosses,acentraltenetofprospecttheory.
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Moneylotteriesandriskaversionp Conceptof“riskaversion”Supposeyoufacethefollowinglottery
10€AB4€
0€ certainty
IfyoupreferBtoA,thenyouare“riskaverse”,sinceforyou,u(4)>0.5*u(10)+0.5*u(0)
100%
50%
50%
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MoneylotteriesandriskaversionForanyoutcomeCi betweenbestandworst,therewillbesomeprobabilitypi,suchthat:
$1000(bestoutcome)Ci ~
$0(worstoutcome)
ForanyCi (e.g.$2,or$10)betweenbestandworstoutcome,therewillbesomeprobabilitypi,suchthatweareindifferentbetweenthecertaintyofCi andthegamblebetweenbestandworst.
100%
p
1-p
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Choiceunderuncertaintyp Expectedutilityprinciple:(u iscardinalandweworkwithuncertainty)
ABEu(A)≥Eu(B)
(Recallthatwithoutuncertaintyu wasordinalandwehadABu(A)≥u(B))
Supposethatanindividualfacesthefollowinglottery(gamble):x1
Whatistheexpectedvalueofthisgamble?x2
! Û
! Û
x~p
1-p
random
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Choiceunderuncertaintyp ,whereisthemean
valueofthegamble(whatIwouldgainonaverage).p Supposethatyoualsohavethechoice
px1 +(1-p)x2
Thegambleandthesurebethaveexactlythesameexpectedvalue.Whichonewouldyouchoose?Theonethatgivesyougreaterutility.So,let’scomparetheutilitylevelsofthegambleandofthesurebet.
xxppxxE =-+= 21 )1()~( x
100%
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Choiceunderuncertaintyp Supposethattheutilityfromx1 isu(x1)andtheutilityfromx2
isu(x2).p Expectedutilityofthegamble:
p Utilityofthesurebet
)()1()()~( 21 xupxpuxEu -+=
))1(())~(()( 21 xppxuxEuxu -+==
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Riskaversionp Wedefineanindividualasrisk-avertifheprefersgainingthe
expectedvaluewithcertaintythanincurringsomeriskbutgainingthesamevalueonaverage.
iffu(px1 +(1-p)x2)≥pu(x1)+(1-p)u(x2)
certainamount gamble
x x~!
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Riskaversionp Wedefineanindividualasrisk-avertifheprefersgainingthe
expectedvaluewithcertaintythanincurringsomeriskbutgainingthesamevalueonaverage.
iffu(px1 +(1-p)x2)≥pu(x1)+(1-p)u(x2)
certainamount gamble
x x~!
Butthishappenswhenu(.)isaconcave function
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Riskaversion
Decreasingmarginalutilityofincome.Theutilitygainfromanextraeuroislowerthantheutilitylossofhavingaeuroless.Henceriskaversion=thefearoflosing
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Risk-lovingattitudep Risk-lovingattitude
iffu(px1 +(1-p)x2)≤pu(x1)+(1-p)u(x2)
certainamount gamble
xx~ !
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Risk-lovingattitudep Risk-lovingattitude
iffu(px1 +(1-p)x2)≥pu(x1)+(1-p)u(x2)
certainamount gamble
x x~!
Butthishappenswhenu(.)isaconvex function
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Risk-lovingattitude
u u(x)
x1xx2Ex
Risk-loving
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Riskneutralityp Wedefineanindividualasrisk-neutral
iffu(px1 +(1-p)x2)=pu(x1)+(1-p)u(x2)
certainamount gamble
x x~~
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Riskneutralityp Wedefineanindividualasrisk-neutral
iffu(px1 +(1-p)x2)=pu(x1)+(1-p)u(x2)
certainamount gamble
x x~~
Thismeansthatu(.)isalinearfunction
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Riskneutrality
u
u(x)
x1x
x2Ex
Risk-neutral
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Examplep SupposethatyoufaceachoicebetweenAandB,where
€10AB€4
€0
IfyoupreferBtoA,canwesaythatyouarerisk-averse?
10.5
0.5
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Examplep SupposethatyoufaceachoicebetweenAandB,where
€10AB€4
€0
IfyoupreferBtoA,canwesaythatyouarerisk-averse?Answer:YES.Becauseforyouu(4)>0.5u(10)+0.5u(0)
10.5
0.5
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Summarysofarp Ifapersonhasaconcaveutilityfunction,heisrisk-averse.Thatisu(px1 +(1-p)x2)≥pu(x1)+(1-p)u(x2)
p Ifapersonhasaconvexutilityfunction,heisarisk-lover.Thatisu(px1 +(1-p)x2)≤pu(x1)+(1-p)u(x2)
p Ifapersonhasalinearutilityfunction,heisarisk-neutralperson.Thatisu(px1 +(1-p)x2)=pu(x1)+(1-p)u(x2)
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Certaintyequivalentp AgainconsiderachoicebetweenAandB.
Risk-averse€20B€10
AB€9€0B€8
Theexpectedvalueofthelotteryis10€.
10.5
0.51
1
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Certaintyequivalentp AgainconsiderachoicebetweenAandB.
Risk-averse€20B€10BA
AB€9€0B€8
10.5
0.51
1 !
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Certaintyequivalentp AgainconsiderachoicebetweenAandB.
Risk-averse€20B€10BA
AB€9BA€0B€8
!1
0.5
0.51
1
!
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Certaintyequivalentp AgainconsiderachoicebetweenAandB.
Risk-averse€20B€10BA
AB€9BA€0B€8B~A
!1
0.5
0.51
1
!
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Certaintyequivalentp AgainconsiderachoicebetweenAandB.
Risk-averse€20B€10BA
AB€9BA€0B€8B~A
!1
0.5
0.51
1
!
Certaintyequivalent (let’sdenoteity):itistheamountthatmakesmeindifferentbetweenthegambleandthesureamount.Itistheamountofmoneythat,ifgainedwithcertainty,providesthesameutilityasthegamble.
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Riskpremiump AgainconsiderachoicebetweenAandB.
Risk-averse€20B€10BA
AB€9BA€0B€8B~A
!1
0.5
0.51
1
!
Riskpremium(π):thedifferencebetweentheexpectedvalueofthegambleandthecertaintyequivalent(:10€-8€=2€).
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Riskpremium(π)p TheriskpremiumisthemoneyIabandoninordertohavemore
safety.
Or,inotherwords,thelossofincomethatcanbeconcededinordertogetridoftherisk(andobtainthecertaintyequivalent).
Itmeasuresthegapbetweentheexpectedvalueofthegambleandthecertaintyequivalent.Itis“positivelycorrelated”withriskaversion.
Insummary, and)~()()( xEuyuxu ==-π xy =+ π
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Graphicalrepresentationu
u(x)
x1xx2
u(x1)
u(x2)
Ex
u(Ex)
y
Eu(x )amount you would sacrifice to eliminate the risk (π)
u(Ex)
Ex= y
§ Expected payoff and the utility of expected payoff.
§ Expected utility and the certainty-equivalent
§ The risk premium
§ Utility values of two payoffs
Eu(x )
•
x
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Graphicalrepresentation(similarfromMWG)
y
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Theprobabilitypremiump Foranyfixedamountofmoneyx andpositivenumberε,the
probabilitypremiumdenotedbyπ(x,ε,u),istheexcessinwinningprobabilityoverfairoddsthatmakestheindividualindifferentbetweenthecertainoutcomex andagamblebetweenthetwooutcomesx+ε andx-ε.Thatis
u(x)=(½+π(x,ε,u))*u(x+ε) +(½- π(x,ε,u))*u(x-ε)
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Theprobabilitypremiumgraphically
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Measuringriskaversionp Howcanwecomparedegreesofriskaversion?p Itmusthavesomethingtodowiththeconcavityoftheutility
function.Moreconcavefunctionsshouldcorrespondtomoreriskaversion.Thehigherthedistancebetweenu(x)andEu(x).
p U’’isameasureofconcavity,butitisnotsuitablebecauseifwelinearlytransformu toau +b,a>0,thesecondderivativeofu isu’’,whilethesecondderivativeofau +b is au’’.
p Solution:standardizewithu’(.)p Butu’(.)willbenegativeforriskaversepersons,soputa
minussigninfrontinordertogetancoefficientofriskaversion.
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TheArrow-Prattmeasureofabsoluteriskaversion
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TheArrow-Prattmeasureofabsoluteriskaversion
Notethat:1.
2.rA (x)isafunctionofx,wherex canbethoughtofastheconsumer’scurrentlevelofwealth.Thuswecanadmitthesituationwheretheconsumerisriskaverse,riskloving,orriskneutralfordifferentlevelsofinitialwealth.
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TheArrow-Prattmeasureofabsoluteriskaversion
3.Wecanalsothinkabouthowthedecisionmaker’sriskaversionchangeswithherwealth.Howdoyouthinkthisshouldgo?Doyoubecomemoreorlesslikelytoacceptagamblethatoffers100withprobability½and−50withprobability½asyourwealthincreases?
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TheArrow-Prattmeasureofabsoluteriskaversion
3.Wecanalsothinkabouthowthedecisionmaker’sriskaversionchangeswithherwealth.Howdoyouthinkthisshouldgo?Doyoubecomemoreorlesslikelytoacceptagamblethatoffers100withprobability½and−50withprobability½asyourwealthincreases?
Hopefully,youansweredmore.Thismeansthatyoubecomelessriskaverseaswealthincreases,andthisishowweusuallythinkofpeople,ashavingnon-increasingabsoluteriskaversion.
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TheArrow-Prattmeasureofabsoluteriskaversion
4.TheAPmeasureiscalledameasureofabsoluteriskaversionbecauseitsayshowyoufeelaboutlotteriesthataredefinedoverabsolutenumbersofdollars.Agamblethatofferstoincreaseordecreaseyourwealthbyacertainpercentageisarelativelottery,sinceitsprizesaredefinedrelativetoyourcurrentlevelofwealth.Wealsohaveameasureofrelativeriskaversion,
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Application:Insurancep Aconsumerhasinitialwealthw.p Withprobabilityπ,theconsumersuffersdamageofD.p Thus,intheabsenceofinsurance,theconsumer’sfinalwealth
isw −D withprobabilityπ,andwwithprobability1−π.p Supposeinsuranceisavailable.Eachunitofinsurancecostsq,
andpays1dollarintheeventofaloss.Supposethepersonbuysα unitsofinsurance.
p Costofinsurance-αq
-αq+α
Supposethattheinsuranceis“actuariallyfair”ifitsexpectedcostiszero.
1-ππ
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Application:Insurancep Exp.Cost=(1-π).(-αq)+π(-αq+α)=0
" q=π(thecosttotheconsumerof1euroofinsuranceisjusttheexpectedcostofprovidingthatcoverage)
Howmanyunitsofinsuranceshouldtheconsumerbuyiftheinsuranceisactuariallyfair?Findα tomaxexpectedutility.
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Application:Insurancew– D– αq+α
w - αq
Maxoverα :Eu =πu(w-D-αq+α)+(1-π)u(w-αq)Firstderivativew.r.t.α:
πu’(w-D-αq+α)(-q+1) +(1-π)u’(w-αq)(-q)=0or u’(w-D-αq+α) =u’(w-αq)
Iftheconsumerisriskaverse,thenu’(.)isstrictlydecreasing,sothat
w-D-αq+α =w-αqOr a*=D (fullinsurance)
π
1-π