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AdvancedMicroeconomicTheory

Chapter5:ChoicesunderUncertainty

Outline

• Simple,Compound,andReducedLotteries• IndependenceAxiom• ExpectedUtilityTheory• MoneyLotteries• RiskAversion• ProspectTheoryandReference-DependentUtility

• ComparisonofPayoffDistributions

AdvancedMicroeconomicTheory 2

Simple,Compound,andReducedLotteries


SimpleLotteries

• Considerasetofpossibleoutcomes(orconsequences)𝐶.

• Theset𝐶 caninclude– simplepayoffs𝐶 ∈ ℝ (positiveornegative)– consumptionbundles𝐶 ∈ ℝ$

• Outcomesarefinite(𝑁 elementsin𝐶,𝑛 =1,2, … ,𝑁)

• Probabilitiesofeveryoutcomeareobjectivelyknown– 𝑝- foroutcome1,𝑝. foroutcome2,etc.


SimpleLotteries

• Simplelottery isalist𝐿 = 𝑝-, 𝑝., … , 𝑝0

with𝑝1 ≥ 0 forall𝑛 and∑ 𝑝1015- = 1,where

𝑝1 isinterpretedastheprobabilityofoutcome𝑛 occuring.

• Insomebooks,lotteriesaredescribedincludingtheoutcomestoo.


SimpleLotteries• Asimplelotterywith2possibleoutcomes

• “Degenerated”probabilitypairs– at(0,1),outcome2happenswithcertainty.

– at(1,0),outcome1happenswithcertainty.

• Strictlypositiveprobabilitypairs– Individualfacessomeuncertainty,i.e.,𝑝- +𝑝. = 1


21 2{ : 1}p R p p= + =

(0,1)

1

1p

2p

1p1

(1,0)

2p

SimpleLotteries• Asimplelotterywith3possibleoutcomes(i.e.,3-dim.simplex).

• Interceptsrepresentdegeneratedprobabilitieswhereoneoutcomeiscertain.

• Pointsstrictlyinsidethehyperplane connectingthethreeinterceptsdenotealotterywheretheindividualfacesuncertainty. AdvancedMicroeconomicTheory 7

1 2 3{ 0 : 1}p p p p= + + =

1

1p1

(0,0,1)

3p

2p1

1p

3p

2p

(1,0,0)(0,1,0)

SimpleLotteries• 2-dim.projectionofthe3-dim.simplex

• Verticesrepresenttheintercepts

• Thedistancefromagivenpointtothesideofthetrianglemeasurestheprobabilitythattheoutcomerepresentedattheoppositevertexoccurs.


1 2 3( , , )L p p p=

3x

1 2

3

1x2x

1 2 3 1where x x x+ + =

SimpleLotteries

• Alotteryliesononeoftheboundariesofthetriangle:–Wecanonlyconstructsegmentsconnectingthelotterytotwooftheoutcomes.

– Theprobabilityassociatedwiththethirdoutcomeiszero.


1 2 1x x+ =3 0x =1 2

3

1x

2x

CompoundLotteries• Givensimplelotteries

𝐿7 = 𝑝-7, 𝑝.7, … , 𝑝07 for𝑘 = 1,2, … , 𝐾andprobabilities𝛼7 ≥ 0 with∑ 𝛼7;

15- = 1,thenthecompoundlottery 𝐿-, 𝐿., … , 𝐿;; 𝛼-, 𝛼., … , 𝛼; istheriskyalternativethatyieldsthesimplelottery𝐿7 withprobability𝛼7 for𝑘 = 1,2, … , 𝐾.– Thinkaboutacompoundlotteryasa“lotteryoflotteries”:first,Ihaveprobability𝛼7 ofplayinglottery1,andifthathappens,Ihaveprobability𝑝-7 ofoutcome1occurring.

– Then,thejointprobabilityofoutcome1is𝑝- = 𝛼- = 𝑝-- + 𝛼. = 𝑝-. + ⋯+ 𝛼; = 𝑝-;


CompoundandReducedLotteries

• Giventhatinterpretation,thefollowingresultshouldcomeatnosurprise:– Foranycompoundlottery𝐿-, 𝐿., … , 𝐿;; 𝛼-, 𝛼., … , 𝛼; ,wecancalculateitscorrespondingreducedlotteryasthesimplelottery𝐿 = 𝑝-, 𝑝., … , 𝑝0 thatgeneratesthesameultimateprobabilitydistributionofoutcomes.

• Thereducedlottery𝐿 ofanycompoundlotterycanbeobtainedby

𝐿 = 𝛼-𝐿- + 𝛼.𝐿. + ⋯+ 𝛼;𝐿; ∈ ∆AdvancedMicroeconomicTheory 11

Reduced Lottery

1 (1,0,0)L =

21 3 3, ,4 8 8

L =

31 3 3, ,4 8 8

L =

11/ 3=

2 1/ 3=

3 1/ 3=1 1 1, ,2 4 4


• Example1:– Allthreelotteriesareequallylikely– P outcome1 = -

H= 1 + -

H= -I+ -

H= -I= -

.

– P outcome2 = -H= 0 + -

H= HJ+ -

H= HJ= -

I

– P outcome3 = -H= 0 + -

H= HJ+ -

H= HJ= -

I


1 2 31 1 1 1 1 1, ,3 3 3 2 4 4

L L L L= + + =

2 3L L=

2

3

2x

1

1 (1,0,0)L =


• Example1(continued):– Probabilitysimplexofthereducedlotteryofacompoundlottery

– Reducedlottery𝐿assignsthesameprobabilityweighttoeachsimplelottery.


1x

Reduced Lottery

41 1, ,02 2

L =

51 1,0,2 2

L =

11/ 2=

2 1/ 2=1 1 1, ,2 4 4

Outcome 1⟶

Outcome 2⟶

1 1 1 1 12 2 2 2 2

+ =

1 1 1 102 2 2 4

+ =Outcome 3⟶

1 1 1 102 2 2 4

+ =


• Example2:– Bothlotteriesareequallylikely



• Example2 (continued):– Probabilitysimplexofthereducedlotteryofacompoundlottery


41 1, ,02 2

L =

2

3

1

(1,0,0)(0,1,0)

(0,0,1)

4 51 1 1 1 1, ,2 2 2 2 4L L L+ = =

51 1,0,2 2

L =


• Consumerisindifferentbetweenthetwocompoundlotterieswhichinducethesamereducedlottery– ThiswasillustratedinthepreviousExamples1and2where,despitefacingdifferentcompoundlotteries,theconsumerobtainedthesamereducedlottery.

• WerefertothisassumptionastheConsequentialisthypothesis:– Onlyconsequences,andtheprobabilityassociatedto

everyconsequence(outcome)matters,butnottheroutethatwefollowinordertoobtainagivenconsequence.


PreferencesoverLotteries

• Foragivensetofoutcomes𝐶,considerthesetofallsimplelotteriesover𝐶,ℒ.

• Weassumethatthedecisionmakerhasacomplete andtransitive preferencerelation≿overlotteriesinℒ,allowinghimtocompareanypairofsimplelotteries𝐿 and𝐿′.– Completeness:Foranytwolotteries𝐿 and𝐿′,either𝐿 ≿ 𝐿′ or𝐿O ≿ 𝐿,orboth.

– Transitivity:Foranythreelotteries𝐿, 𝐿′ and𝐿′′,if𝐿 ≿ 𝐿′ and𝐿O ≿ 𝐿′′,then𝐿 ≿ 𝐿′′.



• Extremepreferenceforcertainty:– 𝐿 ≿ 𝐿′ ifandonlyif

max1∈0

𝑝1 ≥ max1∈0

𝑝1O

– Thedecisionmakerisonlyconcernedabouttheprobabilityassociatedwiththemostlikelyoutcome.



• Smallestsizeofthesupport:– 𝐿 ≿ 𝐿′ ifandonlyif

supp(𝐿) ≤ supp(𝐿′)

wheresupp 𝐿 = 𝑛 ∈ 𝑁:𝑝1> 0 .

– Thedecisionmakerprefersthelotterywhoseprobabilitydistributionisconcentratedoverthesmallestsetofpossibleoutcomes.


PreferencesoverLotteries• Lexicographicpreferences:– First,orderoutcomesfrommostpreferred(outcome1)toleastpreferred(outcome𝑛).

– Then𝐿 ≿ 𝐿′, ifandonlyif𝑝-> 𝑝-O ,or

If𝑝-= 𝑝-O and𝑝.> 𝑝.O ,orIf𝑝-= 𝑝-O and𝑝.= 𝑝.O and𝑝H> 𝑝HO ,or

…– Thedecisionmakerweaklypreferslottery𝐿 to𝐿′ ifoutcome1ismorelikelytooccurinlottery𝐿 thaninlottery𝐿′.

– Ifoutcome1isaslikelytooccurinbothlotteries,hemovestooutcome2;andsoon.AdvancedMicroeconomicTheory 20


• Theworstcasescenario:– First,attachanumber𝑣(𝑧) toeveryoutcome𝑧 ∈𝐶,thatis,𝑣 𝑧 ∈ ℝ.

– Then𝐿 ≿ 𝐿′ ifandonlyif

min 𝑣 𝑧 : 𝑝 𝑧 > 0 > min 𝑣 𝑧 : 𝑝′ 𝑧 > 0

– Thedecisionmakerpreferslottery𝐿 ifthelowestutilityhecangetfromplayinglottery𝐿 ishigherthanthelowestutilityhecanobtainfromplayinglottery𝐿′.



• Continuityofpreferencesoverlotteries:– Continuity1:Foranythreelotteries𝐿, 𝐿′, and𝐿′′,thesets𝛼 ∈ 0,1 : 𝛼𝐿 + 1 − 𝛼 𝐿′ ≿ 𝐿′′ ⊂ [0,1] and𝛼 ∈ 0,1 : 𝐿′′ ≿ 𝛼𝐿 + 1 − 𝛼 𝐿′ ⊂ [0,1]

areclosed.

– Continuity2:if𝐿 ≻ 𝐿′, thenthereisaneighborhoodsof𝐿 and𝐿′,𝐵(𝐿) and𝐵(𝐿′),suchthatforall𝐿c ∈ 𝐵(𝐿) and𝐿d ∈ 𝐵(𝐿′),wehave𝐿c ≻ 𝐿d.



• Smallchangesintheprobabilitydistributionoflotteries𝐿 and𝐿′ donotchangethepreferenceoverthetwolotteries.


( ')B L

1 2

3

aL

bL

( )B L

L

'L

1 2

3

B(L’)


• Example:


If𝐿 ≻ 𝐿′, then𝐿c ≻ 𝐿d.


• Thecontinuityassumption,asinconsumertheory,impliestheexistenceofautilityfunction𝑈: ℒ → ℝ suchthat

𝐿 ≿ 𝐿′ ifandonlyif𝑈(𝐿) ≥ 𝑈(𝐿′)

• However,wefirstimposeanadditionalassumptioninordertohaveamorestructuredutilityfunction.– Thefollowingassumptionisrelatedwithconsequentialism:theIndependenceaxiom.



• IndependenceAxiom(IA):apreferencerelationsatisfiesIAif,foranythreelotteries𝐿,𝐿′,and𝐿′′,and𝛼 ∈ (0,1) wehave

𝐿 ≿ 𝐿′ifandonlyif𝛼𝐿 + 1 − 𝛼 𝐿′′ ≿ 𝛼𝐿′ + 1 − 𝛼 𝐿′′

• Intuition:Ifwemixeachoftwolotteries,𝐿and𝐿′,withathirdone(𝐿′′),thenthepreferenceorderingofthetworesultingcompoundlotteriesisindependentoftheparticularthirdlottery.


2

3

1 2

3

12

3

1 2

3

1


• 𝐿 ≿ 𝐿′ifandonlyif𝛼𝐿 + 1 − 𝛼 𝐿O ≿ 𝛼𝐿′ + (1 − 𝛼)𝐿′′



• Example 1 (intuition):– Thedecisionmakerpreferslottery𝐿 to𝐿O,𝐿 ≿ 𝐿O

– Constructacompoundlotterybyacointoss:§ playlottery𝐿 ifheadscomesup§ playlottery𝐿′′ iftailscomesup

– ByIA,if𝐿 ≿ 𝐿O,then-.𝐿 + -

.𝐿OO ≿ -

.𝐿′ + -

.𝐿′′



• Example2 (violationsofIA):– Extremepreferenceforcertainty– Considertwosimplelotteries𝐿 and𝐿′ forwhich𝐿 ∼ 𝐿O.

– Constructtwocompoundlotteriesforwhich12 𝐿 +

12𝐿 ≁

12 𝐿′ +

12 𝐿

– If𝐿 ∼ 𝐿O,thenitmustbethatmax 𝑝-, 𝑝., … , 𝑝1 = max 𝑝-O , 𝑝.O , … , 𝑝1O



• Example2 (violationsofIA):– Compoundlottery-

.𝐿 + -

.𝐿 coincideswithsimple

lottery𝐿.– Hence,max 𝑝-, 𝑝., … , 𝑝1 isusedtoevaluatelottery𝐿.

– Butcompoundlottery-.𝐿′ + -

.𝐿 isareduced

lotterywithassociatedprobabilities

max12𝑝-

O +12𝑝-, … ,

12 𝑝1

O +12𝑝1

whichmightdiffer frommax 𝑝-O , 𝑝.O , … , 𝑝1O .AdvancedMicroeconomicTheory 30


• Example2 (violationsofIA,anumericalexample):– Considertwosimplelotteries

𝐿 = (0.4, 0.5, 0.1),𝐿 = (0.5, 0, 0.5)– Hence,

max 0.4, 0.5, 0.1 = 0.5 = max 0.5, 0, 0.5implyingthat𝐿 ∼ 𝐿O.

– However,thecompoundlottery-.𝐿′ + -

.𝐿 entails

probabilities0.4 + 0.5

2 ,0.5 + 02 ,

0.1 + 0.52 = 0.45, 0.25, 0.3

implyingthatmax 0.45, 0.25, 0.3 = 0.45.AdvancedMicroeconomicTheory 31


• Example2 (violationsofIA,anumericalexample):– Therefore,

max 0.4, 0.5, 0.1 = 0.5 > 0.45 = max 0.45, 0.25, 0.3

andthus𝐿 = -.𝐿 + -

.𝐿 ≻ -

.𝐿O + -

.𝐿.

– ThisviolatestheIA,whichrequires-.𝐿 + -

.𝐿 ∼ -

.𝐿′ + -

.𝐿



• Example3 (violationsofIA,“worstcasescenario”):– Consider𝐿 ≻ 𝐿O.– Then,thecompoundlottery-

.𝐿 + -

.𝐿 doesnot need

tobepreferredto-.𝐿′ + -

.𝐿.

– Example:§ Considerthesimplelotteries𝐿 = (1,3) and𝐿′ = (10,0),withprobabilities(𝑝-, 𝑝.) and(𝑝-O , 𝑝.O ),respectively.

§ Thisimpliesmin 𝑣 𝑧 : 𝑝 𝑧 > 0 = 1forlottery𝐿min 𝑣 𝑧 : 𝑝′ 𝑧 > 0 = 0 forlottery𝐿′

§ Hence,𝐿 ≻ 𝐿O.AdvancedMicroeconomicTheory 33

PreferencesoverLotteries• Example3 (violationsofIA,“worstcasescenario”):– Example (continued):

§ However,thecompoundlottery-. 𝐿 +-. 𝐿′ is

--. ,

H. ,

whoseworstpossibleoutcomeisH.,whichispreferred

tothatof-. 𝐿 +-. 𝐿,whichis1.

§ Hence,despite𝐿 ≻ 𝐿O oversimplelotteries,

𝐿 = -. 𝐿 +

-. 𝐿 ≺

-. 𝐿 +

-. 𝐿′,

whichviolatestheIA.


ExpectedUtilityTheory



• Theutilityfunction𝑈: ℒ → ℝ hastheexpectedutility(EU)form ifthereisanassignmentofnumbers 𝑢-, 𝑢., … , 𝑢0 tothe𝑁 possibleoutcomessuchthat,foreverysimplelottery𝐿 =𝑝-, 𝑝., … , 𝑝0 ∈ ℒ wehave

𝑈 𝐿 = 𝑝-𝑢- + ⋯+ 𝑝0𝑢0– AutilityfunctionwiththeEUformisalsoreferredtoasavon-Neumann-Morgenstern (vNM)expectedutilityfunction.

– Notethatthisfunctionislinear intheprobabilities.AdvancedMicroeconomicTheory 36

ExpectedUtilityTheory• Hence,autilityfunction𝑈: ℒ → ℝ hastheexpectedutilityformifandonlyifitislinearintheprobabilities,i.e.,

𝑈 n 𝛼7𝐿7;

75-=n 𝛼7 = 𝑈(𝐿7)

;

75-

forany𝐾 lotteries𝐿7 ∈ ℒ,𝑘 = 1,2, … , 𝐾 andprobabilities 𝛼-, 𝛼., … , 𝛼; ≥ 0 and∑ 𝛼7;

75- = 1.

• Intuition:theutilityoftheexpectedvalueofthe𝐾lotteries,𝑈 ∑ 𝛼7𝐿7;

75- ,coincideswiththeexpectedutilityofthe𝐾 lotteries,∑ 𝛼7𝑈(𝐿7);

75- .



• Notethattheutilityoftheexpectedvalueofplayingthe𝐾 lotteriesis

𝑈 n 𝛼7𝐿7;

75-=n 𝑢1 = n𝛼7

�

7

𝑝17�

1

where∑ 𝛼7�7 𝑝17 isthetotaljointprobabilityof

outcome𝑛 occurring.



• Notethattheexpectedutilityfromplayingthe𝐾lotteriesis

n 𝛼7 = 𝑈(𝐿7);

75-=n 𝛼7 = n𝑢1

�

1

𝑝17�

7

where∑ 𝑢1�1 𝑝17 istheexpectedutilityfromplayinga

givenlottery𝑘.



• TheEUpropertyisacardinalproperty:– Notonlyrankmatters,theparticularnumberresultingform𝑈: ℒ → ℝ alsomatters.

• Hence,theEUformispreservedonlyunderincreasinglineartransformations(a.k.a.affinetransformations).– Hence,theexpectedutilityfunction𝑈p: ℒ → ℝ isanothervNM utilityfunctionifandonlyif

𝑈p 𝐿 = 𝛽𝑈 𝐿 + 𝛾

forevery𝐿 ∈ ℒ,where𝛽 > 0.AdvancedMicroeconomicTheory 40

ExpectedUtilityTheory:Representability

• Supposethatthepreferencerelation≿ satisfiesrationality,continuityandindependence.Then,≿admitsautilityrepresentationoftheEUform.

• Thatis,wecanassignanumber𝑢1 toeveryoutcome𝑛 = 1,2, … ,𝑁 insuchamannerthatforanytwolotteries

𝐿 = 𝑝-, 𝑝., … , 𝑝0 and𝐿′ = 𝑝-O , 𝑝.O … , 𝑝0Owehave𝐿 ≿ 𝐿O ifandonlyif𝑈 𝐿 ≥ 𝑈 𝐿′ ,or

n 𝑝1𝑢10

15-≥n 𝑝1O 𝑢1

0

15-• Notation:𝑢1 istheutilitythatthedecisionmakerassignstooutcome𝑛.ItisusuallyreferredastheBernoulliutilityfunction.


ExpectedUtilityTheory:IndifferenceCurves

• LetusnextanalyzetheeffectoftheIAonindifferencecurvesoverlotteries.

1)Indifferencecurvesmustbestraightlines:RecallthatfromtheIA,𝐿~𝐿O impliesthat

𝛼𝐿 + 1 − 𝛼 𝐿$

~𝛼𝐿 + (1 − 𝛼)𝐿O

forall𝛼 ∈ 0,1 .



StraightindifferencecurvesAdvancedMicroeconomicTheory 43

2

3

'L

L

1

If ~ ', then ~ (1 ) 'L L L L L+


• Whyindifferencecurvesmustbestraight?–Wehavethat𝐿~𝐿O,but𝐿 ≺ -

.𝐿 + -

.𝐿O.Thisis

equivalentto12 𝐿 +

12𝐿 ≺

12 𝐿 +

12 𝐿

O

– ButfromtheIAwemusthave12 𝐿 +

12𝐿~

12 𝐿 +

12 𝐿

O

– Hence,indifferencecurvesmustbestraightlinesinordertosatisfytheIA.



• CurvyindifferencecurvesoverlotteriesareincompatiblewiththeIA

– Thecompoundlottery-.𝐿 + -

.𝐿′ wouldnotlieon

thesameindifferencecurveaslottery𝐿 and𝐿′.– Hence,thedecisionmakerisnot indifferentbetweenthecompoundlotteries-

.𝐿 + -

.𝐿′ and

-.𝐿 + -

.𝐿′.



CurvyindifferencecurveAdvancedMicroeconomicTheory 46

2

3

1

L

1 1'2 2L L+

'L


2)Indifferencecurvesmustbeparallellines:Ifwehavethat𝐿~𝐿O,thenbytheIA

13 𝐿 +

23𝐿′′~

13 𝐿′ +

23 𝐿

OO

§ Thatis,theconvexcombinationof𝐿 and𝐿O withathirdlottery𝐿OO shouldalsolieonthesameindifferencecurve.

§ ThisimpliesthattheindifferencecurvesmustbeparallellinesinordertosatisfytheIA.


2

3

1


– Ifcompoundlotteries-H𝐿 + .

H𝐿′′ and-

H𝐿′ +

.H𝐿′′ lieondifferent

(nonparallel)indifferencecurves,then13 𝐿 +

23𝐿′′ ≺

13 𝐿′ +

23 𝐿

OO

whichviolatestheIA.AdvancedMicroeconomicTheory 48

§ NonparallelindifferencecurvesareincompatiblewiththeIA.

ExpectedUtilityTheory:

ViolationsoftheIA:– DespitetheintuitiveappealoftheIA,weencounterseveralsettingsinwhichdecisionmakersviolateit.

–Wenextelaborateontheseviolations.


ExpectedUtilityTheory:ViolationsoftheIA

• Allais’paradox:– Consideralotteryoverthreepossiblemonetaryoutcomes:

– Firstchoiceset:𝐿- = (0,1,0) and𝐿-O = ( -t

-tt, Ju-tt

, --tt)

– Secondchoiceset:𝐿. = (0, --

-tt, Ju-tt) and𝐿.O = ( -t

-tt, 0, ut

-tt)


1st prize 2nd prize 3rd prize

$2.5mln $500,000 $0


– About50%studentssurveyedexpressed𝐿- ≻ 𝐿-O and𝐿.O ≻ 𝐿..

– ThesechoicesviolatetheIA.– Toseethis,considerthatthedecisionmaker’spreferencesoverlotterieshaveaEUform.Hence,𝐿- ≻𝐿-O implies

𝑢v >10100𝑢.v +

89100𝑢v +

1100𝑢t

– BytheIA,wecanadd Ju-tt

𝑢t −Ju-tt

𝑢v onbothsides

𝑢v +89100𝑢t −

89100𝑢v >

10100𝑢.v +

89100𝑢v +

1100𝑢t +

89100𝑢t −

89100𝑢vAdvancedMicroeconomicTheory 51


– Simplifying11100𝑢v +

89100𝑢t

yz{|$}

>10100𝑢.v +

90100𝑢t

yz{|$}~

whichimplies𝐿. ≻ 𝐿.O .

– DidyourownchoicesviolatetheIA?



• ReactionstotheAllais’Paradox:– Approximationtorationality:peopleadapttheirchoicesastheygo.

– Littleeconomicsignificance:thelotteriesinvolveprobabilitiesthatareclosetozeroandone.

– Regrettheory:thereasonwhy𝐿- ≻ 𝐿-O isbecauseIdidn’twanttoregretasurewinof$500,000.

– GiveuptheIAinfavorofaweakerassumption:thebetweennessaxiom.



• Machina’sparadox:– Considerthat

TriptoBarcelona≻ MovieaboutBarcelona≻ Home

– Now,considerthefollowingtwolotteries

𝐿- = ( uu-tt

, --tt

, 0) and 𝐿. = ( uu-tt

, 0, --tt)

– Fromthepreviouspreferencesovercertainoutcomes,howcanweknowthisindividual’spreferencesoverlotteries?§ UsingtheIA.



– From𝑇 ≻ 𝑀 andtheIA,wecanconstructthecompoundlotteries

99100𝑇 +

1100𝑀 ≻

99100𝑀 +

1100𝑀

– From𝑀 ≻ 𝐻 andtheIA,wehave99100𝑀 +

1100𝑀 ≻

99100𝑀 +

1100𝐻

– Bytransitivity,99100𝑇 +

1100𝑀

$�

≻99100𝑇 +

1100𝐻

$}– Hence,𝐿- ≻ 𝐿.. AdvancedMicroeconomicTheory 55


– Therefore,forpreferencesoverlotteriestobeconsistentwiththeIA,weneed𝐿- ≻ 𝐿..

–Manysubjectsinexperimentalsettingswouldratherprefer𝐿.,thusviolatingtheIA.

–Manypeopleexplainchoosing𝐿. over𝐿- ongroundsofthedisappointmenttheywouldexperienceinthecaseoflosingthetriptoBarcelona,andhavingtowatchamovieinstead.§ Similartoregrettheory.



• Dutchbooks:– Intheabovetwoanomalies,actualbehaviorisinconsistentwiththeIA.

– CanwethenrelyontheIA?–WhatwouldhappentoindividualswhosebehaviorviolatestheIA?

– Theywouldbeweededoutofthemarketbecausetheywouldbeopentotheacceptanceofso-calledDutchbooks,leadingthemtoasurelossofmoney.



– Considerthat𝐿 ≻ 𝐿′.BytheIA,weshouldhave𝛼𝐿 + 1 − 𝛼 𝐿

$≻ 𝛼𝐿 + (1 − 𝛼)𝐿O

– If,instead,theIAisviolated,then𝐿 ≺ 𝛼𝐿 + (1 − 𝛼)𝐿O

– Consideranindividualwiththesepreferences,whoinitiallyownslottery𝐿.

– Ifweofferhimthecompoundlottery𝛼𝐿 + (1 −𝛼)𝐿O,forasmallfee$𝑥,hewouldacceptsuchatrade.



– Aftertherealizationstage,heownseither𝐿 or𝐿′§ If𝐿′,thenweoffer𝐿 againfor$𝑦.§ If𝐿,thenweoffer𝛼𝐿 + (1 − 𝛼)𝐿O for$𝑦.

– Eitherway,heisatthesamepositionashestarted(owning𝐿 or𝛼𝐿 + (1 − 𝛼)𝐿O),buthavinglost$𝑥 + $𝑦 intheprocess.

–Wecanrepeatthisprocessadinfinitum.– Hence,individualswithpreferencesthatviolatetheIAwouldbeexploitedbymicroeconomists(theywouldbea“moneypump”).



• Furtherreading:– “Developmentsinnon-expectedutilitytheory:Thehuntforadescriptivetheoryofchoiceunderrisk”(2000)byChrisStarmer,JournalofEconomicLiterature,vol.38(2)

– Choices,ValuesandFrames (2000)byNobelprizewinnersDanielKahneman andAmosTversky,CambridgeUniversityPress.

– TheoryofDecisionunderUncertainty (2009)byItzhak Gilboa,CambridgeUniversityPress.


TheoriesModifyingExpectedUtilityTheory

1)Weightedutilitytheory:– Thepayofffunctionfromplayinglottery𝐿 is

𝑉 𝐿 = n𝑤� = 𝑢(𝑥�)�

�∈�where

𝑤� =� �� (��)

∑ �(��)�(��)��∈�

and𝑔: 𝐶 → ℝ

– Theutilityofoutcome𝑥� ∈ 𝐶 isweightedaccordingto:a) itsprobability𝑝 𝑥�b) outcome𝑥� itselfthroughfunction𝑔: 𝐶 → ℝ



– Example:Consideralotterywithtwopayoffs𝑥- and𝑥.withprobabilities𝑝 and1 − 𝑝. Then,theweightedutilityis

𝑉 𝐿 = 𝑤-𝑢 𝑥- + 𝑤.𝑢 𝑥.=

𝑔 𝑥- 𝑝𝑔 𝑥- 𝑝 + 𝑔 𝑥. 1 − 𝑝 𝑢 𝑥-

+𝑔 𝑥. 1 − 𝑝

𝑔 𝑥- 𝑝 + 𝑔 𝑥. 1 − 𝑝 𝑢(𝑥.)

If𝑔(𝑥�) = 𝑔(𝑥�) forany𝑥� ≠ 𝑥�,then𝑉 𝐿 = 𝑝𝑢 𝑥- + 1 − 𝑝 𝑢 𝑥.

whichisastandardexpectedutilityfunction.AdvancedMicroeconomicTheory 62


• Theweightedutilitytheory reliesonthesameaxiomsasexpectedutilitytheory,exceptfortheIA,whichisrelaxedtothe“weakindependenceaxiom.”–Weakindependenceaxiom:ifwehavethat𝐿-~𝐿., wecanfindapairofprobabilities 𝛼 and 𝛼′suchthat𝛼𝐿- + 1 − 𝛼 𝐿H~𝛼O𝐿. + 1 − 𝛼O 𝐿H

– TheIAbecomesaspecialcaseif𝛼 = 𝛼′.AdvancedMicroeconomicTheory 63


2)Rankdependentutilitytheory:– First,ranktheoutcomes𝑥-, 𝑥., … , 𝑥1 fromworst(𝑥-)tobest(𝑥1)

– Second,applyaprobabilityweightingfunction𝑤� = 𝜋 𝑝� + ⋯+ 𝑝1 − 𝜋 𝑝��- + ⋯+ 𝑝1

𝑤1 = 𝜋 𝑝1where𝜋(⋅) isanon-decreasingtransformationfunction,with𝜋(0) = 0 and𝜋(1) = 1.

– Finally,arank-dependentutilityis

𝑉 𝐿 = n𝑤� ⋅ 𝑢(𝑥�)�

�∈�AdvancedMicroeconomicTheory 64


– Foralotterywithtwooutcomes,𝑥- and𝑥. where𝑥. > 𝑥-,therank-dependentutilityis

𝑉(𝐿) = 𝑤(𝑝)𝑢(𝑥-) + (1 − 𝑤(𝑝))𝑢(𝑥.)

where𝑝 istheprobabilityofoutcome𝑥-.

– Thismodelallowsfordifferentweighttobeattachedtoeachoutcome,asopposedtoexpectedutilitytheorymodelsinwhichthesameutilityweightisattachedtoalloutcomes.



– Transformationfunction𝜋(⋅)


π(p)

π(p)=p

π(p)>p1

0 1p

π(p)

π(p)=p

π(p)<p

1

0 1p

Pessimisticπ(p) Optimisticπ(p)

π(p)

1

0 1pπ(p)=p


• EmpiricalevidencesuggestsanS-shapedtransformationfunction.

• Intuition:individualsarepessimisticinrareoutcomes(i.e.,𝑝 < 𝑝),butbecomeoptimisticforoutcomestheyhavefrequentlyencountered.



• Therank-dependentutilitytheoryreliesonthesameaxiomsasexpectedutilitytheory,exceptfortheIA,whichisreplacedbyco-monotonicindependence.


MoneyLotteries


MoneyLotteries

• Wenowrestrictourattentiontolotteriesovermonetary amounts,i.e.,𝐶 = ℝ.

• Moneyiscontinuousvariable,𝑥 ∈ ℝ,withcumulativedistributionfunction(CDF)

𝐹 𝑥 = 𝑃𝑟𝑜𝑏 𝑦 ≤ 𝑥 forall𝑦 ∈ ℝ


11/2x

45o

F(.)

1/2

1

F(x)=xUniform

Distribution

MoneyLotteries

• Auniform,continuousCDF,𝐹 𝑥 = 𝑥– Sameprobabilityweighttoeverypossiblepayoff


11/2x

F(.)

1/2

1

MoneyLotteries

• Anon-uniform,continuousCDF,𝐹 𝑥


1 2 x

F(.)

1/2

1

1/4

3/4

3 4 5 6 7...

MoneyLotteries

• Anon-uniform,discreteCDF

𝐹 𝑥 =

0if𝑥 < 114 if𝑥 ∈ [1, 4)34 if𝑥 ∈ [4, 6)

1if𝑥 ≥ 6


f(.)

x

MoneyLotteries

• If𝑓 𝑥 isadensityfunctionassociatedwiththecontinuous CDF𝐹 𝑥 ,then

𝐹 𝑥 = � 𝑓 𝑡 𝑑𝑡�

¡¢


1 2

f(.)

1/2

1/4

3 4 5 6 7 x

MoneyLotteries

• If𝑓 𝑥 isadensityfunctionassociatedwiththediscreteCDF𝐹 𝑥 ,then

𝐹 𝑥 =n𝑓 𝑡�

£¤�


MoneyLotteries

• Wecanrepresentsimplelotteriesby𝐹 𝑥 .• Forcompoundlotteries:– IfthelistofCDF’s𝐹- 𝑥 ,𝐹. 𝑥 ,...,𝐹; 𝑥represent𝐾 simplelotteries,eachoccurringwithprobability𝛼-, 𝛼., … , 𝛼; ,thenthecompoundlotterycanberepresentedas

𝐹 𝑥 =n 𝛼7𝐹7 𝑥;

75-

– Forsimplicity,assumethatCDF’saredistributedovernon-negativeamountsofmoney.


MoneyLotteries

–WecanexpressEUas

𝐸𝑈 𝐹 = ∫𝑢 𝑥 𝑓 𝑥 𝑑𝑥�� or∫𝑢 𝑥 𝑑𝐹(𝑥)�

�

where𝑢 𝑥 isanassignmentofutilityvaluetoeverynon-negativeamountofmoney.– Ifthereisadensityfunction𝑓 𝑥 associatedwiththeCDF𝐹(𝑥),thenwecanuseeitheroftheexpressions.Ifthereisno,wecanonlyusethelatter.

– Note:wedonotneedtowritedownthelimitsofintegration,sincetheintegralisoverthefullrangeofpossiblerealizationsof𝑥.


MoneyLotteries

–𝐸𝑈 𝐹 isthemathematicalexpectationofthevaluesof𝑢 𝑥 ,overallpossiblevaluesof𝑥.–𝐸𝑈 𝐹 islinearintheprobabilities

§ Inthediscreteprobabilitydistribution,𝐸𝑈 𝐹 = 𝑝- 𝑢- + 𝑝. 𝑢. + ⋯

– TheEUrepresentationissensitivenotonlytothemean ofthedistribution,butalsotothevariance,andhigherordermomentsofthedistributionofmonetarypayoffs.§ Letusnext analyzethisproperty.


MoneyLotteries• Example:Letusshowthatif𝑢 𝑥 = 𝛽𝑥. + 𝛾𝑥,thenEUisdeterminedbythemeanandthevariancealone.– Indeed,

𝐸𝑈 𝑥 = �𝑢 𝑥 𝑑𝐹 𝑥�

�

= � 𝛽𝑥. + 𝛾𝑥 𝑑𝐹 𝑥�

�

= 𝛽�𝑥.𝑑𝐹 𝑥�

�§ �}

+ 𝛾�𝑥𝑑𝐹 𝑥�

�§ �

– Ontheotherhand,weknowthat𝑉𝑎𝑟 𝑥 = 𝐸 𝑥. − 𝐸 𝑥 . ⟹𝐸 𝑥. = 𝑉𝑎𝑟 𝑥 + 𝐸 𝑥 .


MoneyLotteries

• Example (continued):– Substituting𝐸 𝑥. in𝐸𝑈 𝑥 ,

𝐸𝑈 𝑥 = 𝛽𝑉𝑎𝑟 𝑥 + 𝛽 𝐸 𝑥 .

ª§ �}+ 𝛾𝐸 𝑥

– Hence,theEUisdeterminedbythemeanandthevariancealone.


MoneyLotteries

• Recallthatwereferto𝑢 𝑥 astheBernoulliutilityfunction,while𝐸𝑈 𝑥 isthevNMfunction.

• Weimposedfewassumptionson𝑢 𝑥 :– Increasinginmoneyandcontinuous

• Wemustimposeanadditionalassumption:– 𝑢 𝑥 isbounded– Otherwise,wecanendupinrelativelyabsurdsituations(St.Petersburg-Menger paradox).


MoneyLotteries

• St.Petersburg-Menger paradox:– ConsideranunboundedBernoulliutilityfunction,𝑢 𝑥 .Then,wecanalwaysfindanamountofmoney𝑥« suchthat𝑢 𝑥« > 2«,foreveryinteger𝑚.

– Nowconsideralotteryinwhichwetossacoinrepeatedlyuntiltailscomeup.Wegiveamonetarypayoffof𝑥« iftailsisobtainedatthe𝑚th toss.

– Theprobabilitythattailscomesupinthem-thtossis-

.= -.= -.=. . 𝑚times = -

..


MoneyLotteries– Then,theEUofthislotteryis

𝐸𝑈(𝑥) =n12« 𝑢(𝑥«)

¢

«5-– But,becauseof𝑢 𝑥« > 2«,wehavethat

𝐸𝑈 𝑥 =n12« 𝑢 𝑥«

¢

«5-≥n

12« 2

«¢

«5-

=n 1 =¢

«5-+∞

whichimpliesthatthisindividualwouldbewillingtopayinfiniteamountsofmoneytobeabletopaythislottery.

– Hence,weassumethattheBernouilli utilityfunctionisbounded.


MeasuringRiskPreferences


MeasuringRiskPreferences• Anindividualexhibitsriskaversionif

�𝑢 𝑥 𝑑𝐹 𝑥�

�

≤ 𝑢 �𝑥𝑑𝐹 𝑥�

�foranylottery𝐹(=)

• Intuition:– Theutilityofreceivingtheexpectedmonetaryvalueofplaying

thelottery(left-handside)ishigherthan…– Theexpectedutilityfromplayingthelottery(right-handside).

• Ifthisrelationshiphappenswitha) =,wedenotethisindividualasriskneutralb) <,wedenotehimasriskaverterc) ≥,wedenotehimasrisklover.



• Graphicalillustration:– Consideralotterywithtwoequallylikelyoutcomes,$1and$3,withassociatedutilitiesof𝑢(1) and𝑢(3),respectively.

– Expectedvalueofthelottery is𝐸𝑉 = -.= 1 + -

.= 3 = 2,with

associatedutilityof𝑢(2).

– Expectedutilityofthelotteryis-.𝑢 1 + -

.𝑢(3).



• Riskaverseindividual– Utilityfromtheexpectedvalueofthelottery,𝑢(2),ishigher thantheEUfromplayingthelottery,-.𝑢 1 + -

.𝑢(3).

AdvancedMicroeconomicTheory 871 2

u(x)

3

u(3)

u(2)

u(1)

x

1 1(1) (3)2 2u u+

u(x)


• Riskneutralindividual– Utilityfromtheexpectedvalueofthelottery,𝑢(2),coincides withtheEUofplayingthelottery,-.𝑢 1 + -

.𝑢(3).

AdvancedMicroeconomicTheory 881 2

u(x)

3

u(3)

u(1)

u(x)

x

1 1(1) (3) (2)2 2u u u+ =


• Risklovingindividual– Utilityfromtheexpectedvalueofthelottery,𝑢(2),islower thantheEUfromplayingthelottery,-.𝑢 1 + -

.𝑢(3).


u(x)

1 2 3 x

u(x)

(3)u1 1(1) (3)2 2u u+

(2)u(1)u


• Certaintyequivalent,𝑐(𝐹, 𝑢):– Analternativemeasureofriskaversion– Itistheamountofmoneythatmakestheindividualindifferentbetweenplayingthelottery𝐹(=),andacceptingacertainamount𝑐(𝐹, 𝑢).Thatis,

𝑢 𝑐(𝐹, 𝑢) = ∫𝑢 𝑥 𝑑𝐹 𝑥�� or∑𝑢 𝑥 𝑓 𝑥�

�

– 𝑐(𝐹, 𝑢) isbelow(above)theexpectedvalueofthelotteryforriskaverse(lover)individuals,andexactlycoincidesforriskneutralindividuals.



– 𝑐(𝐹, 𝑢) istheamountofmoney(𝑥)forwhichutilityisequaltotheEUofthelottery𝑢 𝑐(𝐹, 𝑢) =

12𝑢 1 +

12𝑢(3)

– Riskpremium(RP):theamountthatarisk-aversepersonwouldpay toavoidtakingarisk:

𝑅𝑃 = 𝐸𝑉 − 𝑐 𝐹, 𝑢 > 0AdvancedMicroeconomicTheory 91

• Certaintyequivalentforarisk-averseindividual

1 2

u(x)

3

u(3)

u(2)

u(1)

x

c(F,u), Certainty Equivalent

1 1(1) (3)2 2u u+

1.87

Risk premium

u(x)


– Individualwouldhavetobegivenanamountofmoneyabove theexpectedvalueofthelotteryinordertoconvincehimto“stopplaying”thelottery:𝑅𝑃 = 𝐸𝑉 − 𝑐 𝐹, 𝑢 < 0


• Certaintyequivalentforarisklover

1 2

u(x)

3

u(3)

u(2)

u(1)

u(x)

x

c(F,u)RiskPremium

1 1(1) (3)2 2u u+


– Thecertaintyequivalent𝑐 𝐹, 𝑢 coincideswiththeexpectedvalueofthelottery.

– Hence,𝑅𝑃 = 𝐸𝑉 − 𝑐 𝐹, 𝑢 = 0


• Certaintyequivalentforariskneutralindividual

u(x)

1 2 3 x

u(x)(3)u

1 1(1) (3) (2)2 2u u u+ =

(1)u

c(F,u)


• Probabilitypremium, 𝜋(𝑥, 𝜀, 𝑢):– Analternativemeasureofriskaversion– Itistheexcessinwinningprobabilityoverfairoddsthatmakestheindividualindifferentbetweenthecertaintyoutcome𝑥 andagamblebetweenthetwooutcomes𝑥 + 𝜀 and𝑥 − 𝜀:

𝑢 𝑥

=12 + 𝜋 𝑥, 𝜀, 𝑢 𝑢 𝑥 + 𝜀 +

12 − 𝜋 𝑥, 𝜀, 𝑢 𝑢 𝑥 − 𝜀

– Intuition:Betterthanfairoddsmustbegivenfortheindividualtoaccepttherisk.


MeasuringRiskPreferences• The“extraprobability”𝜋 thatisneededtomaketheEUofthelotterycoincideswiththeutilityoftheexpectedlottery:

𝑢 2 =12 + 𝜋 𝑢 3 +

12 − 𝜋 𝑢 1



• Thefollowingpropertiesareequivalent:1) Thedecisionmakerisriskaverse.2) TheBernoulliutilityfunction𝑢 𝑥 isconcave,

𝑢OO(𝑥) ≤ 0.3) Thecertaintyequivalentislowerthanthe

expectedvalueofthelottery,i.e.,𝑐(𝐹, 𝑢) ≤∫𝑢 𝑥 𝑑𝐹 𝑥�� .

4) Theriskpremiumispositive,𝑅𝑃 = 𝐸𝑉 − 𝑐 𝐹, 𝑢 .5) Theprobabilitypremiumispositiveforall𝑥 and𝜀,

i.e.,𝜋(𝑥, 𝜀, 𝑢) ≥ 0.AdvancedMicroeconomicTheory 96

MeasuringRiskPreferences• Arrow-Prattcoefficientofabsoluteriskaversion:

𝑟³ 𝑥 = −𝑢OO(𝑥)𝑢O(𝑥)

– Clearly,thegreaterthecurvatureoftheutilityfunction,𝑢OO(𝑥),thelargerthecoefficient𝑟³ 𝑥 .

– But,whydonotwesimplyhave𝑟³ 𝑥 = 𝑢OO(𝑥)?• Becauseitwillnotbeinvarianttopositivelineartransformationsoftheutilityfunction,suchas𝑣 𝑥 = 𝛽𝑢 𝑥 .Thatis,𝑣′′ 𝑥 = 𝛽𝑢′′ 𝑥 isaffectedbythetransformation,buttheabovecoefficientofriskaversionisunaffected.

𝑟³ 𝑥 = −𝛽𝑢OO(𝑥)𝛽𝑢O(𝑥) = −

𝑢OO(𝑥)𝑢O(𝑥)



• Example (CARAutilityfunction).– Take𝑢 𝑥 = −𝑒¡c� where𝑎 > 0.Then

𝑟³ 𝑥 = −𝑢OO(𝑥)𝑢O(𝑥) = −

−𝑎.𝑒¡c�

𝑎𝑒¡c� = 𝑎

whichisconstantinwealth𝑥.– TheliteraturereferstothisBernoulliutilityfunctionastheConstantAbsoluteRiskAversion(CARA).


MeasuringRiskPreferences• If𝑟³ 𝑥 decreasesasweincreasewealth𝑥,thenwesaythatsuchBernoulliutilityfunctionsatisfiesdecreasingabsoluteriskaversion(DARA)

𝜕𝑟³ 𝑥𝜕𝑥 < 0

• Intuition:wealthierpeoplearewillingtobearmoreriskthanpoorerpeople.Note,however,thatthisisNOTduetodifferentutilityfunctions,butbecausethesameutilityfunctionisevaluatedathigher/lowerwealthlevels.

• AsufficientconditionforDARAis𝑢OOO 𝑥 > 0.



• Arrow-Prattcoefficientofrelativeriskaversion:

𝑟¶ 𝑥 = −𝑥 = ·~~(�)·~(�)

or𝑟¶ 𝑥 = 𝑥 = 𝑟³ 𝑥

– 𝑟¶ 𝑥 doesnotvarywiththewealthlevelatwhichitisevaluated.

– Wecanshowthat𝜕𝑟¶ 𝑥𝜕𝑥 = 𝑟³ 𝑥

�+ 𝑥 =

𝜕𝑟³ 𝑥𝜕𝑥

– Therefore,𝜕𝑟¶ 𝑥𝜕𝑥 < 0⇒⇍

𝜕𝑟³ 𝑥𝜕𝑥 < 0



• Example:– Take𝑢 𝑥 = 𝑥d.Then

𝑟¶ 𝑥 = −𝑥 =𝑏 𝑏 − 1 𝑥d¡.

𝑏𝑥d¡- = 1 − 𝑏

forall𝑥.– TheliteraturereferstothisBernoulliutilityfunctionastheConstantRelativeRiskAversion(CRRA).



• Example (continued):– ConsideraCRRAutilityfunction𝑢 𝑥 = 𝑥d for𝑏 =1, -., -H, -I.

– 𝑟¶ 𝑥 increases,respectively,to-

., .H, HI,

makingutilityfunctionmoreconcave.


0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

x

Money, x

Utility

x1/2x1/3

x1/4Increasing degree of risk aversion

MeasuringRiskPreferences• Autilityfunction𝑢³(=) exhibitsmorestrongriskaversionthananotherutilityfunction𝑢º(=) if,thereisaconstant𝜆 > 0,

𝑢³OO(𝑥-)𝑢ºOO(𝑥-)

≥ 𝜆 ≥𝑢³O (𝑥.)𝑢ºO (𝑥.)

• Inaddition,if𝑥- = 𝑥.,theaboveconditioncanbere-writtenas

𝑢³OO(𝑥-)𝑢³O (𝑥-)

≥𝑢ºOO(𝑥-)𝑢ºO (𝑥-)

• Then,𝑢³(=) alsoexhibitsmoreriskaversionthan𝑢º(=).



• Fortwoutilityfunctions𝑢- and𝑢.,where𝑢.isaconcavetransformationof𝑢-,thefollowingpropertiesareequivalent:1) Thereexistsanincreasingconcavefunction𝜑(=)

suchthat𝑢. 𝑥 = 𝜑(𝑢-(𝑥)) forany𝑥.Thatis,𝑢. = ismoreconcavethan𝑢- = .

2) 𝑟³ 𝑥, 𝑢. ≥ 𝑟³ 𝑥, 𝑢- forany𝑥.3) 𝑐(𝐹, 𝑢.) ≤ 𝑐(𝐹, 𝑢-) foranylottery𝐹(=).4) 𝜋(𝑥, 𝜀, 𝑢.) ≥ 𝜋(𝑥, 𝜀, 𝑢-) forany𝑥 and𝜀.



5) Whenever𝑢. = findsalottery𝐹(=) atleastasgoodasarisklessoutcome�̅�,then𝑢- = alsofindssuchalottery𝐹(=) atleastasgoodas�̅�.Thatis

𝐸𝑈. = �𝑢. 𝑥 𝑑𝐹 𝑥�

�

≥ 𝑢. �̅� ⟹

𝐸𝑈- = �𝑢- 𝑥 𝑑𝐹 𝑥�

�

≥ 𝑢- �̅�


MeasuringRiskPreferences• Differentdegreesofriskaversion

• 𝑢- = and𝑢. = areevaluatedatthesamewealthlevel𝑥.

• Thesamelotteryyieldsalargerexpectedutilityfortheindividualwithlessriskaverse preferences,𝐸𝑈- > 𝐸𝑈..

• 𝑐 𝐹, 𝑢. < 𝑐(𝐹, 𝑢-),reflectingthatindividual2ismoreriskaverse.


1 x

u(x)

3 x

2( , )c F u1( , )c F u

EU1

EU2

u1(x)=u2(x)

u1(x)

u2(x)

ProspectTheoryandReference-DependentUtility


ProspectTheory

• Prospecttheory:adecisionmaker’stotalvaluefromalistofpossibleoutcomes 𝑥 = (𝑥-, 𝑥., … , 𝑥1) withassociatedprobabilities𝑝 = (𝑝-, 𝑝., … , 𝑝1) is

𝑣 𝑥, 𝑝 =n 𝑤(𝑝�) = 𝑣(𝑥�)1

�5-

where– 𝑤(𝑝�) isa“probabilityweightingfunction”– 𝑣 𝑥� isthe“valuefunction”theindividualobtainsfromoutcome𝑥�


ProspectTheory

• Threemaindifferencesrelativetostandardexpectedutilitytheory:

• First,𝑤 𝑝� ≠ 𝑝�:– if𝑤 𝑝� > 𝑝�,individualsoverestimate thelikelihoodofoutcome𝑥�

– if𝑤 𝑝� < 𝑝�,individualsunderestimate thelikelihoodofoutcome𝑥�

– if𝑤 𝑝� = 𝑝�,themodelcoincideswithstandardexpectedutilitytheory.


ProspectTheory

• Second,everypayoff𝑥� isevaluatedrelativetoa“referencepoint”𝑥t,withthevaluefunction𝑣 𝑥� ,whichis– Increasingandconcave,𝑣OO 𝑥� < 0,forall𝑥� > 𝑥t,• Thatis,theindividualisriskaverseforgains.

– Decreasingandconvex,𝑣OO 𝑥� > 0,forall𝑥� < 𝑥t• Thatis,theindividualisriskloverforlosses.

– Extremes:• if𝑥t = 0,theindividualisriskaverseforallpayoffs;• if𝑥t = +∞,heisriskloverforallpayoffs.


ProspectTheory

• Third,valuefunction𝑣 𝑥� hasakinkatthereferencepoint𝑥t.– Thecurvebecomessteeperforlosses(totheleftof𝑥t)thanforgains(totherightof𝑥t).• Lossaversion:• Agivenlossof$aproducesalargerdisutilitythanagainofthesameamount.


ProspectTheory• Valuefunctioninprospecttheory


ProspectTheory


• Example:– ConsiderasinTversky andKahneman (1992)

𝑤 𝑝 = �¾

�¾�(-¡�)¾�¾

and𝑣 𝑥 = 𝑥¿

where0 < 𝛽 < 1,and0 < 𝛼 < 1.• Notethatthisimpliesprobabilityweighting,

butdoesnotconsideravaluefunctionwithlossaversionrelativetoareferencepoint.

ProspectTheory• Example (continued):– Depictingtheprobabilityweightingfunction


ProspectTheory


• Example:– Acommonvaluefunctionis

𝑣 𝑥� = 𝑥�¿ if𝑥� ≥ 𝑥t,and= −𝜆(−𝑥�)¿ if𝑥� < 𝑥t

where0 < 𝛼 ≤ 1,and𝜆 ≥ 1 representslossaversion.• If𝜆 = 1 theindividualdoesnotexhibitloss

aversion.

ProspectTheory


• Example:• Commonsimplifications,assume𝛼 = 𝛽 = 1

(whichimpliesnoprobabilityweighting,andlinearvaluefunctions),toestimate𝜆.

• Averageestimates𝜆 = 2.25 and𝛽 = 0.88

ProspectTheory• Furtherreading:– NicholasBarberis (2013)“ThirtyYearsofProspectTheoryinEconomics:AReviewandAssessment,”JournalofEconomicPerspectives,27(1),pp.173-96.

– R.DuncanLuceandPeterC.Fishburn (1991)“Rankandsign-dependentlinearutilitymodelsforbinarygambles.”JournalofEconomicTheory,53,pp.75–100.

– DanielKahneman andAmosTversky (1992)“Advancesinprospecttheory:Cumulativerepresentationofuncertainty”JournalofRiskandUncertainty,5(4),pp.297–323.

– PeterWakker andAmosTversky (1993)“Anaxiomatization ofcumulativeprospecttheory.”JournalofRiskandUncertainty,7,pp.147–176.


Reference-DependentUtility

• Individualpreferencesareaffectedbyreferencepoints.Thus,gainsandlosescanbeevaluateddifferently.

• Consideraconsumptionvector𝑥 ∈ ℝ1 whichisevaluatedagainsta𝑛-dimensionalreferencevector𝑟 ∈ ℝ1.Utilityfunctionis

𝑢 𝑥 𝑟 = 𝑚 𝑥 + 𝑛(𝑥|𝑟)

where𝑛 𝑥7 𝑟7 = 𝜇 𝑚7 𝑥7 − 𝑚7(𝑟7)measuresthegain/lossofconsuming𝑥7 unitsofgood𝑘 relativetoitsreferenceamount𝑟7.



• Forlotterieswithcumulativedistributionfunction𝐹(𝑥),

𝑈 𝐹 𝑟 = ∫ 𝑢 𝑥 𝑟 𝑑𝐹(𝑥)

• Forlotteriesoverthesetofreferencepoints

𝑢 𝐹 𝐺 = ∫ ∫ 𝑢 𝑥 𝑟 𝑑𝐺(𝑟)𝑑𝐹(𝑥)



• Furtherreading:– “Reference-DependentConsumptionPlans”(2009)byKoszegi andRabin,AmericanEconomicReview,vol.99(3).

– “RationalChoicewithStatusQuoBias”(2005)byMasatlioglu andOk,JournalofEconomicTheory,vol.121(1).

– “Onthecomplexityofrationalizingbehavior”(2007) Apesteguia andBallester,EconomicsWorkingPapers1048.


ComparisonofPayoffDistributions



• Sofarwecomparedutilityfunctions,butnotthedistributionofpayoffs.

• Twomainideas:1) 𝐹(=) yieldsunambiguouslyhigherreturnsthan

𝐺(=).Wewillexplorethisideainthedefinitionoffirstorderstochasticdominance(FOSD);

2) 𝐹(=) isunambiguouslylessriskythan𝐺(=).Wewillexplorethisideainthedefinitionofsecondorderstochasticdominance(SOSD).


ComparisonofPayoffDistributions• FOSD:𝐹(=) FOSD𝐺(=) if,foreverynon-decreasingfunction𝑢:ℝ → ℝ,wehave


�

≥ �𝑢 𝑥 𝑑𝐺 𝑥�

�• Thedistributionofmonetarypayoffs𝐹(=) FOSDthedistributionofmonetarypayoffs𝐺(=) ifandonlyif

𝐹(𝑥) ≤ 𝐺 𝑥 or1 − 𝐹 𝑥 ≥ 1 − 𝐺 𝑥forevery𝑥.

• Intuition:Foreveryamountofmoney𝑥,theprobabilityofgettingatleast𝑥 ishigherunder𝐹(=) thanunder𝐺(=).



• Atanygivenoutcome𝑥,theprobabilityofobtainingprizesabove𝑥 ishigherwithlottery𝐹(=)thanwithlottery𝐺(=),i.e.,1 − 𝐹 𝑥 ≥ 1 − 𝐺 𝑥 .


F(x)

1

xx

G(x)( )G x

( )F x

1()

Fx

1()

Gx

$1 $2 $3 $4 $5 Dollars12

12

14

12

14

0

0 0

0

0G(∙)

F(∙)


• Example:– Letustakelotteries𝐹(=) and𝐺(=) overdiscreteoutcomes.


Howcanweknowif𝐹(=) FOSD𝐺(=)?

ComparisonofPayoffDistributions• Example(continued):– 𝐹(=) liesbelowlottery𝐺 = .Hence,𝐹(=) concentratesmoreprobabilityweightonhighermonetaryoutcomes.

– Thus,𝐹(=) FOSD𝐺(=).

AdvancedMicroeconomicTheory 126$1 $2

F(x)

1/2

1/4

$3 $4 $5

3/4

1

F(.)

G(.)

1/4

1/41/2

x

1/2

ComparisonofPayoffDistributions• Example(Binomialdistribution):

– Considerthebinomialdistribution

𝐹 𝑥;𝑁, 𝑝 =𝑁𝑝 𝑝�(1 − 𝑝)0¡�

– where𝑥 ∈ 0,𝑁 . Assuming𝑁 = 100 andparameter𝑝 increasingfrom𝑝 = -I to𝑝 =

-. .

Then,𝐹 𝑥; 100,1/2 FOSD𝐹 𝑥; 100,1/4 .


10060 804020

1.0

0.8

0.6

0.4

0.212

p =14

p =


• Wenowfocusontheriskiness ordispersion ofalottery,asopposedtohigher/lowerreturnsoflottery(FOSD).

• Tofocusonriskiness,weassumethattheCDFswecomparehavethesamemean (i.e.,sameexpectedreturn).

• SOSD:𝐹(=) SOSD𝐺(=) if,foreverynon-decreasingfunction𝑢:ℝ → ℝ,wehave


�

≥ �𝑢 𝑥 𝑑𝐺 𝑥�

�AdvancedMicroeconomicTheory 128

$1 $2 $3 $4 $5 Dollars12

12

14

14

14

0 0

0

0G(∙)

F(∙) 14

ComparisonofPayoffDistributions• Example (Mean-PreservingSpread):– Letustakelotteries𝐹(=) and𝐺(=) overdiscreteoutcomes.– Lottery𝐺(=) spreadstheprobabilityweightoflottery𝐹(=)overalargersetofmonetaryoutcomes.

– Themeanisnonethelessunaltered(2.5).– Forthesetworeasons,wesaythataCDFisameanpreservingspreadoftheother.


𝐹(=)

𝐺(=)


• 𝐺(=) isamean-preservingspreadof𝐹(=),butitisriskierthan𝐹(=) intheSOSDsense.

• NotethatneitherFOSDtheother– 𝐹(=) isnotabove/below𝐺(=) forall𝑥


$1 $2

F(.)

1/2

1/4

$3 $4 $5

3/4

1F(.)

G(.)

Dollars

12

12


• Example (Elementaryincreaseinrisk):– 𝐺(=)isanElementaryIncreaseinRisk(EIR)ofanotherCDF𝐹(=)if𝐺(=)takesalltheprobabilityweightofaninterval 𝑥O, 𝑥OO andtransfersittotheendpointsofthisinterval,𝑥O and𝑥OO,suchthatthemeanoftheoriginallotteryispreserved.

– EIRisamean-preservingspread(MPS),buttheconverseisnotnecessarilytrue:

𝐸𝐼𝑅⇒⇍𝑀𝑃𝑆– Hence,if𝐺(=) isanEIRof𝐹(=),then𝐹(=) SOSD𝐺(=).



• Example (continued):– bothCDFs𝐹(=)and𝐺(=)maintainthesamemean.

– 𝐺(=)concentratesmoreprobabilityattheendpointsoftheinterval 𝑥O, 𝑥OO than𝐹(=).


F(x), G(x)

1

'x ''x

Areas of same size

G(x)F(x)

x


• Hazardratedominance:Thehazardrateoflottery𝐹(𝑥) is

𝐻𝑅Æ(𝑥) =𝑓(𝑥)

1 − 𝐹(𝑥)– Intuition:Itmeasurestheinstantaneousprobabilityofaneventhappeningattime𝑥 giventhatitdidnothappenbefore𝑥.

– Example:acomputerstopsworkingatexactly𝑥– If𝐻𝑅Æ(𝑥) ≤ 𝐻𝑅Ç(𝑥),lottery𝐹 𝑥 dominates𝐺 𝑥 intermsofthehazardrate.


ComparisonofPayoffDistributions– Since−𝐻𝑅Æ(𝑥) canbeexpressedas

−𝐻𝑅Æ 𝑥 =𝑑𝑑𝑥 ln 1 − 𝐹(𝑥)

– Solvingfor𝐹(𝑥),

𝐹 𝑥 = 1 − exp −� 𝐻𝑅Æ 𝑡 𝑑𝑡�

t– Then,

𝐹 𝑥 = 1 − exp −� 𝐻𝑅Æ 𝑡 𝑑𝑡�

t

≤ 1 − exp −� 𝐻𝑅Æ 𝑡 𝑑𝑡�

t= 𝐺 𝑥

– Thus,𝐻𝑅Æ(𝑥) ≤ 𝐻𝑅Ç(𝑥) impliesthat𝐹 𝑥 FOSD𝐺 𝑥 .AdvancedMicroeconomicTheory 134


• Reversehazardrate:Thereversehazardrateoflottery𝐹(𝑥) is

𝑅𝐻𝑅Æ(𝑥) =𝑓(𝑥)𝐹(𝑥)

– Intuition:Itmeasurestheprobabilitythat,conditionalontherealizedpayoffinthelotterybeingequalorlowerthan𝑥,thepayoffyoureceiveisexactly𝑥.

– If𝑅𝐻𝑅Æ(𝑥) ≥ 𝑅𝐻𝑅Ç(𝑥),lottery𝐹 𝑥 dominates𝐺 𝑥 intermsofthereversehazardsense.



– Letusexpress𝑅𝐻𝑅Æ(𝑥) as

𝑅𝐻𝑅Æ 𝑥 =𝑑𝑑𝑥 ln 𝐹(𝑥)

– Solvingfor𝐹(𝑥),

𝐹 𝑥 = exp −� 𝑅𝐻𝑅Æ 𝑡 𝑑𝑡�

t

– Then,

𝐹 𝑥 = exp −� 𝑅𝐻𝑅Æ 𝑡 𝑑𝑡�

t≤ exp −� 𝑅𝐻𝑅Æ 𝑡 𝑑𝑡

�

t= 𝐺 𝑥

– Thus,𝑅𝐻𝑅Æ(𝑥) ≥ 𝑅𝐻𝑅Ç(𝑥) impliesthat𝐹 𝑥 FOSD𝐺 𝑥 .



• Likelihoodratio:Thelikelihoodratioofalottery𝐹 𝑥 is

𝐿𝑅Æ =𝑓(𝑦)𝑓(𝑥)

foranytwopayoffs𝑥 and𝑦,where𝑦 > 𝑥.– 𝐹 𝑥 dominates𝐺 𝑥 intermsoflikelihoodratioif

𝑓(𝑥)𝑔(𝑥) ≤

𝑓(𝑦)𝑔(𝑦)



• 𝐿𝑅 dominanceimplies𝐻𝑅 dominance:– Letusrewrite𝐿𝑅 dominanceas

𝑔(𝑦)𝑔(𝑥) ≤

𝑓(𝑦)𝑓(𝑥)

– Then,forall𝑥,�

𝑔(𝑦)𝑔(𝑥) 𝑑𝑦 ≤ �

𝑓(𝑦)𝑓(𝑥) 𝑑𝑦

¢

t

¢

t

– Simplifying-¡Ç(�)�(�)

≤ -¡Æ(�)É(�)

or É(�)-¡Æ(�)

≤ �(�)-¡Ç(�)

whichimplies𝐻𝑅Æ 𝑥 ≤ 𝐻𝑅Ç 𝑥 .AdvancedMicroeconomicTheory 138


• Summary:– 𝐿𝑅 dominanceimplies𝐻𝑅 dominance– 𝐻𝑅 and𝑅𝐻𝑅 dominanceimplyFOSD.


Appendix5.1:State-DependentUtility


State-DependentUtility

• Sofarthedecisionmakeronlycaredaboutthepayoffarisingfromeveryoutcomeofthelottery.

• Nowweassumethatthedecisionmakercaresnotonlyabouthismonetaryoutcomes,butalsoaboutthestateofnature thatcauseseveryoutcome.– Thatis,𝑢ÊËÌËÍ-(𝑥) ≠ 𝑢ÊËÌËÍ.(𝑥) forgiven𝑥.



• Letusassumethateachofthepossiblemonetarypayoffsinalotteryisgeneratedbyanunderlyingcause(i.e.,anunderlyingstateofnature).

• Examples:– Themonetarypayoffofaninsurancepolicyisgeneratedbyacaraccident§ Stateofnature={caraccident,nocaraccident}

– Themonetarypayoffofacorporatestockisgeneratedbythestateoftheeconomy§ Stateofnature={economicgrowth,economicdepression} AdvancedMicroeconomicTheory 142


• Generally,let𝑠 ∈ 𝑆 denoteastateofnature,where𝑆 isafiniteset.

• Everystate𝑠 hasawell-defined,objectiveprobability𝜋Ï ≥ 0.

• Arandomvariableisfunction𝑔: 𝑆 → ℝ,thatmapsstatesintomonetarypayoffs.



• Examples (revisited):– Caraccident:therandomvariableassignsamonetaryvaluetothestateofnaturecareaccident,andtothestateofnaturenoaccident.


Stateofnature Probability Monetary payoffCaraccident 𝜋ÌÐÐÑÒÍÓË Damage +Deductible– Premium=$1,000Nocaraccident 𝜋Ó{ÌÐÐÑÒÍÓË Premium=-$50


• Examples (revisited):– Corporatestock:therandomvariableassignsamonetaryvaluetothestateofnatureecon.growth,andtothestateofnatureeco.depression.


Stateofnature Probability Monetary payoffEconomic growth 𝜋ÔÕ{ÖË× Dividends, higherpriceofshares=$250

Economic depression 𝜋ÒÍØÕÍÊÊÑ{Ó Nodividends,lossifwesellshares = -$125

State-DependentUtility• Everyrandomvariable𝑔(=) canbeusedtorepresentlottery𝐹(=) overmonetarypayoffsas

𝐹 𝑥 = n 𝜋Ï

�

Ï:�(Ï)Ù�where{𝑠: 𝑔(𝑠) ≤ 𝑥} representsallthosestatesofnature𝑠 thatgenerateamonetarypayoff𝑔(𝑠) ∈ ℝbelowacutoffpayoff𝑥.

• Therandomvariable𝑔(=) generatesamonetarypayoffforeverystateofnature𝑠 ∈ 𝑆,andsince𝑆 isfinite,wecanrepresentthislistofmonetarypayoffsas

(𝑥-, 𝑥., … , 𝑥Ü) ∈ ℝ�Üwhere𝑥Ï isthemonetarypayoffcorrespondingtostateofnature𝑠.



• Example:– Arandomvariable𝑔(=)describesthemonetaryoutcomeassociatedtothefourstatesofnature𝑆 ={1,2,3,4}.

– Outcomesareorderedfromlowertohigher𝑥- ≤ 𝑥. ≤ 𝑥H ≤ 𝑥I.


Prob.

1

1x 3x g(s)2x 4x

1/4

1/2

3/4112

=

214

=

3 0=

414

=


• Example (continued):– Hence,

𝐹(𝑥-) = 𝜋- =-.

𝐹(𝑥.) = 𝜋- + 𝜋. =-.+ -

I= H

I

𝐹(𝑥H) = 𝜋- + 𝜋. + 𝜋H =-.+ -

I+ 0 = H

I𝐹(𝑥I) = 𝜋- + 𝜋. + 𝜋H + 𝜋I = 1

• Disadvantageof𝐹(𝑥):– Foragiven𝑥,wecannotkeeptrackofwhichstate(s)ofnaturegenerated𝑥.


State-DependentUtility:ExtendedEUrepresentation

• Wenowhaveapreferencerelation≿ rankslistsofmonetarypayoffs(𝑥-, 𝑥., … , 𝑥Ü) ∈ ℝ�Ü .

• Notethesimilarityofthissettingwiththatinconsumertheory:– Preferencesoverbundlesthen,preferencesoverlistsofmonetarypayoffshere.

– Since(𝑥-, 𝑥., … , 𝑥Ü) ∈ ℝ�Ü specifiesonepayoffforeachstateofnature,thislistisreferredtoascontingentcommodities.



• Preferencerelation≿ hasanExtendedEUrepresentation ifforevery𝑠 ∈ 𝑆,thereisafunction𝑢Ü:ℝ� → ℝ�Ü (mappingthemonetaryoutcomeofstate𝑠,𝑥Ï,intoautilityvalueinℝ),suchthatforanytwolistsofmonetaryoutcomes(𝑥-, 𝑥., … , 𝑥Ü) ∈ ℝ�Ü and(𝑥-O , 𝑥.O , … , 𝑥ÜO) ∈ ℝ�Ü ,

(𝑥-, 𝑥., … , 𝑥Ü) ≿ (𝑥-O , 𝑥.O , … , 𝑥ÜO) iff

n𝜋Ï𝑢Ï(𝑥Ï)�

Ï

≥ n𝜋Ï𝑢Ï(𝑥ÏO)�

Ï• ThemaindifferencewiththeprevioussectionsisthatnowtheBernoulliutilityfunctionisstate-dependent,𝑢Ï(=),whereasintheprevioussectionsitwasstate-independent,𝑢(=).



• Graphicalrepresentation:– First,atthe“certaintyline”thedecisionmakerreceivesthesamemonetaryamount,regardlessthestateofnature,𝑥- = 𝑥..

– Second,allthe(𝑥-, 𝑥.) pairsonagivenind. curvesatisfy𝜋- = 𝑢- 𝑥- + 𝜋. = 𝑢. 𝑥. = 𝑈Ý

– Third,theuppercontoursetofanind. curvethatpassesthroughpoint(�̅�-, �̅�.) satisfy

𝜋- = 𝑢- 𝑥- + 𝜋. = 𝑢. 𝑥.≥ 𝜋- = 𝑢- �̅�- + 𝜋. = 𝑢. �̅�.

or,moregenerally,∑ 𝜋Ï𝑢Ï(𝑥Ï)�Ï ≥ ∑ 𝜋Ï𝑢Ï(�̅�Ï)�

Ï .AdvancedMicroeconomicTheory 151


• Graphicalrepresentation:– Fourth,movementalongagivenind.curvedoesnotchangethedecisionmaker’sutilitylevel.Hence,totallydifferentiating

𝜋- =𝜕𝑢- �̅�-𝜕𝑥-

𝑑𝑥- + 𝜋. =𝜕𝑢. �̅�.𝜕𝑥.

𝑑𝑥. = 0

andre-arranging,

𝑑𝑥.𝑑𝑥-

= −𝜋- =

𝜕𝑢- �̅�-𝜕𝑥-

𝜋. =𝜕𝑢. �̅�.𝜕𝑥.

= −𝜋- = 𝑢-O (�̅�-)𝜋. = 𝑢.O (�̅�.)

whichrepresentstheslopeoftheind.curve,evaluatedatpoint(�̅�-, �̅�.).ThisisreallysimilartoMRS. AdvancedMicroeconomicTheory 152


– Theslopeoftheind.curveat(�̅�-, �̅�.) is𝑑𝑥.𝑑𝑥-

= −𝜋- = 𝑢-O (�̅�-)𝜋. = 𝑢.O (�̅�.)

– IftheBernoulliutilityisstate-independent,i.e.,𝑢- = = 𝑢. = = ⋯ =𝑢Ü = ,thentheslopeis

𝑑𝑥.𝑑𝑥-

= −𝜋-𝜋. AdvancedMicroeconomicTheory 153

1x

2x 45 line(certainty line)o

1 2x x=

1x

2x

1 2

1 1 1 2 2 2

1 1 1 2 2 2

( , ) such that( ) ( )

( ) ( )

x xu x u xu x u x

++

1 1 1 2 2 2( ) ( )u x u x u+ =

• Graphicalrepresentation:


• Example (Insurancewithstate-dependentutility):– Startfromaninitialsituationof 𝑤,𝑤 − 𝐷withoutinsurance,where𝐷 islossfromaccident.

– Afterinsuranceispurchased,thedecisionmakergetsapaymentof𝑧- instate1,and𝑧. instate2,where𝑧- ≶ 0 and𝑧. ≶ 0,

𝑤 + 𝑧-, 𝑤 − 𝐷 + 𝑧.–Moreover,ifthepolicyisactuariallyfair,thenitsexpectedpayoffiszero,

𝜋-𝑧- + 𝜋.𝑧. = 0AdvancedMicroeconomicTheory 154

1x

2x 1 2x x=

w

w D

1

2

slope =

(Accident)

(No Accident)

1 2 1 1 2 2{( , ) : 0}w z w D z z z+ + + =


• Example (continued):– Thebudgetlineis𝑧. = −à�

à}𝑧-



• Withoutstatedependency:– Indifferencecurvesaretangenttothebudgetlineatthecertaintyline,sincetheslopeoftheindifferencecurveis−à�

à}.

– Hence,thedecisionmakerwouldinsurecompletelysincehisconsumptionlevelisunaffectedbythepossibilityofsufferinganaccident.



• Withstatedependency:– IndifferencecurvesareNOTtangenttothebudgetlineatthecertaintyline.

• Example (continued):– Thedecision-makerprefersapointsuchas(𝑥-O , 𝑥.O ) tothecertainoutcome(�̅�, �̅�).

– Thatis,at(�̅�, �̅�) heprefershigherpayoffsinstate1thaninstate2if𝑢-O �̅� > 𝑢.O (�̅�).Otherwise,hewouldpreferhigherpayoffsinstate2thaninstate1.


1x

2x1 2x x=

1

2

slope =

Slope of Ind. Curve at

(No Accident)

' '1 2( , )x x

1 2( , )x x

1 2( , )x x

1u2u

45 line(certainty line),o


– Notethat𝑢-O �̅� > 𝑢.O (�̅�) impliesthat·�~ �̅

·}~ �̅> 1

and−à�⋅·�~ �̅à}⋅·}~ �̅

< − à�à}.



• Letusnowallowforthepossibilitythatthemonetarypayoffunderstate𝑠,𝑥Ï,isnotacertainamountofmoney,butarandomamountwithdistributionfunction𝐹Ï(⋅).

• Hence,allmonetaryoutcomesarisingfromthe𝑆statesofworldcanbedescribedasalottery𝐿 =(𝐹-, 𝐹., … , 𝐹Ü).

• Giventhis“extended”definitionoflotteries,wecanthenre-writetheIA,asthe“extended”IA.



• ExtendedIA:ThepreferencerelationsatisfiestheextendedIAif,foranythreelotteries𝐿,𝐿O,and𝐿OO and𝛼 ∈ (0,1),wehavethat

𝐿 ≿ 𝐿O iff𝛼𝐿 + (1 − 𝛼)𝐿OO ≿ 𝛼𝐿O + (1 − 𝛼)𝐿OO

• Hence,“extended”IAisamereextensionofthestandardIAtothecaseof“extended”lotteries𝐿 = (𝐹-, 𝐹., … , 𝐹Ü).



• ExtendedEUtheorem:SupposepreferencesrelationsatisfiescontinuityandtheextendedIA.Thenwecanassignautilityfunction𝑢Ï(=)formoneyineverystate𝑠 suchthatforanytwolotteries𝐿 = (𝐹-, 𝐹., … , 𝐹Ü) and 𝐿O =(𝐹-O, 𝐹.O, … , 𝐹ÜO) wehave

𝐿 ≿ 𝐿O iff

n �𝑢Ï 𝑥Ï 𝑑𝐹Ï(𝑥Ï)�

�

�

Ï

≥ n �𝑢Ï 𝑥Ï 𝑑𝐹ÏO(𝑥Ï)�

�

�

ÏAdvancedMicroeconomicTheory 161

Appendix5.2:SubjectiveProbabilityTheory


SubjectiveProbabilityTheory

• Sofarwewereassumingthatprobabilitieswereobjectiveandobservable.

• Thisisnotthecaseincertaincases.Insteadpeoplemightholdprobabilisticbeliefs aboutthelikelihoodofacertainevent:subjectiveprobability.



• Canwededucesubjectiveprobabilityfromactualbehavior?Yes!

• Imagineadecisionmakerwhoprefersagamble

($1instate1,$0instate2) ≿($0instate1,$1instate2)

• Ifthevalueofmoneyisthesameacrossstates,thenhemustbeassigningahighersubjectiveprobabilitytostate1thantostate2.



• Letusstartwithsomedefinitions.• First,wedefinestate𝑠 preferences,≿Ï,onstate𝑠 lotteries𝐹Ï(=) by𝐹Ï(=) ≿ 𝐹ÏO(=) if

�𝑢Ï 𝑥Ï 𝑑𝐹Ï(𝑥Ï)�

�

≥ �𝑢Ï 𝑥Ï 𝑑𝐹ÏO(𝑥Ï)�

�• Hence,thestatepreferences(≿-, ≿., … , ≿Ü)onstatelotteries(𝐹-, 𝐹., … , 𝐹Ü) arestateuniformif

≿Ï=≿Ï~ foranytwostates𝑠 and𝑠�′AdvancedMicroeconomicTheory 165


• Thatis,preferencesoverlotteriesarestateuniformifforanytwostates𝑠 and𝑠�′,therankingofanytwolotteries𝐹Ï(=) and𝐹ÏO(=)coincidesinbothstates,i.e.,

𝐹Ï(=) ≿ 𝐹ÏO(=) or𝐹ÏO(=) ≿ 𝐹Ï(=) or𝐹Ï = ~𝐹ÏO(=)



• Withstateuniformity,𝑢Ï = and𝑢Ï~ = candifferonlyuptoanincreasinglineartransformation.

• Thatis,thereisautilityfunction𝑢 = suchthat𝑢Ï = = 𝜋Ï𝑢 = + 𝛽Ï𝑢Ï~ = = 𝜋Ï~𝑢 = + 𝛽Ï~

foreverystate𝑠 and𝑠O,andforevery𝜋Ï, 𝜋Ï~ > 0and𝛽Ï, 𝛽Ï~ > 0.

• Inwords,therankingbetweentheexpectedutilityofstate𝑠 and𝑠O remainsunaffected.



• SubjectiveprobabilitiesEUtheorem:– SupposethatapreferencerelationsatisfiescontinuityandtheextendedIA,andthatpreferencesoverlotteriesarestateuniform.

– Then,therearesubjectiveprobabilities(𝜋-, 𝜋., … , 𝜋Ü) ≫ 0 andautilityfunction𝑢 = oncertainamountsofmoney,suchthatforanytwolistsofmonetaryamounts(𝑥-, 𝑥., … , 𝑥Ü)and(𝑥-O , 𝑥.O , … , 𝑥ÜO),

(𝑥-, 𝑥., … , 𝑥Ü) ≿ (𝑥-O , 𝑥.O , … , 𝑥ÜO) iff

n𝜋Ï𝑢Ï(𝑥Ï)�

Ï

≥ n𝜋Ï𝑢Ï(𝑥ÏO)�

ÏAdvancedMicroeconomicTheory 168


• Intuition:adecisionmakerprefersthefirstlistofmonetaryoutcomestothesecondifthe“subjective”expectedutilityfromthefirstlistislargerthanthatfromthesecond.

• ThepredictionsofthesubjectiveEUtheoremarenotnecessarilysatisfiedinallexperimentalsettings.– Example:Ellsbergparadox


SubjectiveProbabilityTheory• Ellsbergparadox:– Anurncontains300balls:100areredandtheremaining200areeitherblueorgreen.

– Wefirstpresentthefollowingtwogamblestoagroupofstudents,askingeachofthemtochooseeithergambleAorB.§ GambleA:$1000iftheballisred§ GambleB:$1000iftheballisblue

– Wenextpresentthefollowingtwogamblestothesamegroupofstudents,askingeachofthemtochooseeithergambleCorD.§ GambleC:$1000iftheballisnotred§ GambleD:$1000iftheballisnotblue


SubjectiveProbabilityTheory• Ellsbergparadox (continued):– Commonchoices:peoplechooseAtoB,andCtoD.– ButthesechoicesviolatesubjectiveEUtheory!– Weknowthat

𝑝 Red = 1 − 𝑝(notRed)𝑝 Blue = 1 − 𝑝(notBlue)

– IfgambleAispreferredtoB,thenwemusthave𝑝 Red 𝑢 $1000 > 𝑝 Blue 𝑢 $1000 ⟹

𝑝 Red > 𝑝 Blue– AndifgambleCispreferredtoD,thenwemusthave

𝑝 notRed 𝑢 $1000 > 𝑝 notBlue 𝑢 $1000 ⟹𝑝 notRed > 𝑝 notBlue

– Buttheabovetwoexpressionsareincompatible.AdvancedMicroeconomicTheory 171

Appendix5.3:AmbiguityandAmbiguity

Aversion


AmbiguityandAmbiguityAversion

• AlternativetheoriesthataccountfortheanomalyintheEllsbergparadox:1) expectedutilitytheorywithmultiplepriors(also

referredtoasmaxmin expectedutility)2) rank-dependentexpectedutility(orChoquet

expectedutility)• Individualshaveambiguous(unclear)beliefs,ratherthanobjectiveorsubjectivebeliefs.

• Let𝑓 denoteanact𝑓: 𝑠 → 𝑥 fromthesetofstatestothesetofoutcomes.



• Maxmin expectedutility (MEU):– Ifsubjectshavetoolittleinformationtoformtheirpriors,onecouldalternativelyallowthemtoconsiderasetofpriors.

– Ifanindividualisuncertaintyaverse,hewillchooselottery𝑓 overanotherlottery𝑔 iftheformerprovidesahigherexpectedutilitythanthelatteraccordingtohisworstpossibleprior.



• Uncertaintyaversion:Consideranindividualwhoisindifferentbetweentwolotteries𝑓 and𝑔.Then,heisuncertaintyaverseifheweaklyprefersthecompoundlottery𝛼𝑓 + 1 − 𝛼 𝑔tolottery𝑓,where𝛼 ∈ (0,1).– Intuition:adecisionmakerwhoisuncertaintyaversehasapreferenceformixing(orhedging),sincethecompoundlotterybecomesatleastasvaluableaseitherofthetwolotteriesalone.



• Certainty-independence:Foranytwolotteries𝑓 and𝑔 andaconstantact𝑘 (i.e.,acertainoutcomeoralotterythatremainsconstantacrossallstates),thedecisionmakerweaklypreferslottery𝑓 to𝑔 ifandonlyifheprefers𝛼𝑓 + 1 − 𝛼 𝑘 to𝛼𝑔 + 1 − 𝛼 𝑘 ,where𝛼 ∈(0,1).– Certainty-independenceaxiomrelaxestheIAasitonlyrequiresthatpreferencesovertwolotteriestobeunaffectedwheneachlotteryismixedwithacertainoutcome𝑘.



– Adecisionmakerweaklypreferslottery𝑓 to𝑔 ifandonlyif

min�∈�

� 𝑢 𝑓 𝑠 𝑑𝑝 𝑠 ≥ min�∈�

�𝑢 𝑔 𝑠 𝑑𝑝(𝑠)�

Ü

�

Ü

– Thatis,theindividualevaluatestheexpectedutilityoflotteries𝑓 and𝑔 accordingtoeachofhismultiplepriors𝑝 ∈ 𝐶,andthenselectsthelotterythatyieldsthehighestoftheworstpossibleexpectedutilities.



• Example:– ConsideradecisionmakerwithBernoulliutilityfunction𝑢 𝑥 = 𝑥� ,where𝑥 ≥ 0 denotesmonetaryamounts.

– Assumethatthedecisionmakerfacestwolotteries𝐿³ = $1, $100𝐿º = ($3, $5)

– Also,assumethatthedecisionmaker’spriorsare(𝑝³, 1 − 𝑝³) for𝐿³(𝑝º, 1 − 𝑝º) for𝐿ºAdvancedMicroeconomicTheory 178

AmbiguityandAmbiguityAversion• Example(continued):– AccordingtoMEU,thedecisionmakerchooseslottery𝐿º if

min�æ[𝑝º 3� + (1 − 𝑝º) 5� ]

≥ min�ç[𝑝³ 1� + (1 − 𝑝³) 100� ]

– Ifthedecisionmakerdoesnothaveanyavailableinformationwithwhichtoupdatehispriors,priorscantakevalues(𝑝³, 𝑝º) ∈ (0,1).

– Itispossiblethatinhismostpessimisticbelief,hereceivesthelowestmonetaryamountwithprobabilityone.



• Example(continued):– Then,withargmin 𝑝º = 1,

min�æ[𝑝º 3� + (1 − 𝑝º) 5� ] = 3�

– Similarly,withargmin 𝑝³ = 1,min�ç[𝑝³ 1� + (1 − 𝑝³) 100� ] = 1�

– HenceadecisionmakerwithMEUpreferencesselectslotter𝐿º because 3� ≥ 1� .



• Choquet expectedutility (CEU):– Definebeliefswiththeuseofcapacities.– Acapacityisdefinedasareal-valuedfunction𝑣(=)fromasubsetofthestatespace𝑆 to[0,1],withthenormalization𝑣 ∅ = 0 and𝑣 𝑆 = 1.

– Ifthecapacity𝑣(=) satisfiesmonotonicity,𝑣(𝐴) ≥𝑣(𝐵),where𝐴 isasupersetof𝐵.

–Wecannotuseastandardintegraloverstatessincethecapacity𝑣(=) doesnotcorrespondtoournotionofbeliefs.



– Adecisionmakerweaklyprefers𝑓 to𝑔 iftheChoquet integralssatisfy

� 𝑢 𝑓 𝑆 𝑑𝑣 𝑆 ≥ �𝑢(𝑔 𝑆 )𝑑𝑣(𝑆)�

Ü

�

Ü

– TheCEUandMEUmodelsareconnectedifweimposetheuncertaintyaversionaxiominCEUcontext.Forthatweneedthatcapacity 𝑣(=)satisfiessupermodularity,i.e.,

𝑣 𝐴 ∪ 𝐵 − 𝑣(𝐵) ≥ 𝑣 𝐴 ∪ 𝐶 − 𝑣(𝐶)where𝐶 isasubsetof𝐵,i.e.,𝐶 ⊂ 𝐵.



• Example:– WhiletheuseofChoquet integralsisinvolved,theliteratureoftenuses“simple”capacities.

– Asimplecapacityonstatespace𝑆 canbeunderstoodasaconvexcombinationbetweentwoextremecapacities:

1. astandardprobabilityweighton𝐴,𝑝 𝐴 ∈ 0,1 .2. the“completeignorance”capacity𝑤,where𝑤 𝑆 = 1 and𝑤 𝐴 = 0 forevery𝐴 ⊆ 𝑆.



• Example(continued):– Formally,simplecapacitiesaredefinedas

𝑣 𝐴 = 𝜆𝑝 𝐴 + 1 − 𝜆 𝑤(𝐴)forevery𝐴 ⊆ 𝑆 andwhere𝜆 ∈ 0,1 .

– Parameter𝜆 denotestheindividual’sdegreeofconfidenceon𝑝 𝐴 ,while(1 − 𝜆) captureshisdegreeofambiguityabout𝑝 𝐴 .

– Forfurtherreading,seeHaller(2000)andAflaki (2013).


AmbiguityandAmbiguityAversion• Furtherreading:

– Choquet,G.(1953).Theoryofcapacities.Ann.Inst.Fourier (Grenoble)5131-295.

– Dow,J.andS.Werlang.(1992).Uncertaintyaversion,riskaversion,andtheoptimalchoiceofportfolio.Econometrica, (1),197.

– Epstein,L.andT.Wang.(1994).Intertemporal AssetPricingunderKnightianUncertainty. Econometrica, (2),283-322.

– Hansen,L.,Sargent,T.(2001).“RobustControlandModelUncertainty”.AmericanEconomicReview91,60-66.

– Machina,M.(2014). Handbookoftheeconomicsofriskanduncertainty (Firstedition).Elsevier.

– Mukerji,S.andJ.Tallon (2004).Ambiguityaversionandtheabsenceofwageindexation. JournalofMonetaryEconomics, (3),653-670.

– Nishimura,K.andH.Ozaki.(2004).Searchandknightian uncertainty. JournalofEconomicTheory, (2),299-333.

– Schmeidler,D.(1989).SubjectiveProbabilityandExpectedUtilityWithoutAdditivity.Econometrica,57,571-587.

– Uppal,R.andT.Wang.(2003).ModelMisspecificationandUnderdiversification. JournalofFinance, (6),2465-2486.


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