lecture 5 2013 choice under uncertainty · 2016. 12. 3. · lecture 5 microeconomic theroy consumer...

LECTURE 5MICROECONOMIC THEROY

CONSUMER THEORYChoice under Uncertainty

(MWG chapter 6, sections A-C, and Cowell chapter 8)

Lecturer: Andreas Papandreou

Introductionp Contents

n Expectedutilitytheoryn Measuresofriskaversionn Measuresofrisk

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?

IntroductionEXAMPLEp Supposeweareconsideringtwodifferentuncertainalternatives,eachofwhichoffersadifferentdistributionoverthreeoutcomes:n IbuyyouatriptoBermudan youpayme$500n youdoallmyundergraduatetutorialsofmicro

Introductionp TheprobabilityofeachoutcomeunderalternativesAandB

arethefollowing:

p Wewouldliketoexpressyourutilityforthesetwoalternativesintermsoftheutilityyouassigntoeachindividualoutcomeandtheprobabilitythattheyoccur

Bermuda -$500 Do micro tutorials

A 0.3 0.4 0.3B 0.2 0.7 0.1

Introductionp Supposeyouassign:

n ValueuB tothetriptoBermudan Valueum topayingmethemoneyn Valueuttodoingthetutorials

p Andweknow:n ProbabilitypB tothetriptoBermudan ProbabilitypB topayingmethemoneyn Probabilitypt todoingthetutorials

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

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)

ExpectedUtilityPrinciplep Theonlydifferenceisthatwemaximizeexpectedutility.p Inch.3ofMWGwebasedouranalysisontheassumption

thataconsumerhasrationalpreferencesp However,theassumptionofrationalpreferencesover

uncertainoutcomesisnotsufficienttorepresentthesepreferencesbyautilityfunctionthathastheexpectedutilityform

p Tobeabletodoso,wehavetoplaceadditionalstructureonpreferences

p Thenweshowhowutilityfunctionsoftheexpectedutilityformcanbeusedtostudybehaviorunderuncertainty,anddrawtestableimplications

Assumptionsonpreferences:1.Rationality

1.Theindividualhascompleteandtransitivepreferencesoverdifferentoutcomes(rationality)n ForanyCi,Cj Ci Cj orCj Ci(orboth)

Assumptionsonpreferences2.Reductionofcompoundlotteries

2.Reductionofcompoundlotteries(orconsequentialistpreferences)

Alotteryisaprobabilitydistributionoverasetofpossibleoutcomes.Asimplelottery isavectorL=(p1,p2,…,pN)suchthatpn≥0foralln andΣnpn=1.

Inacompoundlottery anoutcomemayitselfbeasimplelottery

AssumptionsonpreferencesReductionofcompoundlotteries

p Reductionofcompoundlotteries(consequentialistpreferences):Consumerscareonlyaboutthedistributionoverfinaloutcomes,notwhetherthisdistributioncomesaboutasaresultofasimplelottery,oracompoundlottery.Inotherwords,theconsumerisindifferentbetweenanytwocompoundlotteriesthatcanbereducedtothesamesimplelottery.Thispropertyisoftencalledreductionofcompoundlotteries.

p Becauseofthereductionproperty,wecanconfineourattentiontothesetofallsimplelotteries.

AssumptionsonpreferencesReductionofcompoundlotteries

A ~ BC3 C3

p Anotherwaytothinkaboutthereductionpropertyisthatwe’reassumingthereisnoprocess-orientedutility.Consumersdonotenjoytheprocessofthegamble,onlytheoutcome,eliminatingthe“funofthegamble”insettingslikecasinos.

p Onlytheoutcomematters,nottheprocess.

p1p1·p2

p1·p3

Assumptionsonpreferences3.Theindependenceaxiom

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

Assumptionsonpreferences4.Continuity

SupposeC1 isthebestoutcomeSupposeCn istheworstoutcome

ForanyoutcomeCi betweenbestandworst,therewillbesomeprobabilitypi,suchthatweareindifferentbetween:

C1 $1000Ci ~

Cn $0$1000

e.g.$400~$0

100%pi

1-pipi

Assumptionsonpreferencesp Theaboveassumptionscombinedmeanthatanylotterymaybewrittenasasimplerlotterythatonlyinvolvesthebestandtheworstoutcome.

p Example:

Assumptionsonpreferences5.Monotonicity

C1(best) C1 (best)

A BCm(worst) Cm (worst)

ABiffp ≥q

1-p 1-q

ExpectedUtilityTheoremp Theexpectedutilitytheoremsaysthatifaconsumer’s

preferencesoversimplelotteriesarerational,continuous,andexhibitthereductionandindependenceproperties,thenthereisautilityfunctionoftheexpectedutilityformthatrepresentsthosepreferences.

Thatis,therearenumbersu1,...,uN suchthat

nnupLEU )(

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

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)

ExpectedUtilityTheorem:examplep AmIallowedtouseordinalutilityfunctionswhenIworkwithuncertainty?

u(noumbrella,sunny)=100=u1u(umbrella,rains)=8 =u2u(umbrella,sunny)=6 =u3u(noumbrella,rains)=0 =u4

Thisisstillautilityfunctionpreservingtheorderofpreferences(ordinal)

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)

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.

Von-Neumann-MorgensternUF(vN-M)p Inthepreviousexample

Couldbetransformedtou(noumbrella,sunny)=100=v1u(umbrella,rains)=80 =v2u(umbrella,sunny)=60 =v3u(noumbrella,rains)=0 =v4

Wherev(.)=α*u(.)+βwithα=10 andβ=0

Von-Neumann-MorgensternUF(vN-M)p Inthepreviousexample

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

Von-Neumann-MorgensternUF(vN-M)p ThenumbersassignedbythevN-Mutilityfunctionhavecardinalsignificance.

p Supposeu(A)=30u(B)=20u(C)=10

IsAthreetimesbetterthanC?Answer:NO(a>0)

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.

Von-Neumann-MorgensternUF(vN-M)

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.

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

Von-Neumann-MorgensternUF(vN-M)-Example

p Choosejusttwoutilitylevels(α u(.)+ β):givesustwodegreesoffreedom.SoUA =1andUD=0,thensolveforUB andUC

Objectionswiththetheoryofexpectedutility:Allaisparadox

p Allais(1953):presentparticipantswithtwodifferentexperiments.

p Firstexperiment:choosebetweenAandB

$2,500A$2,400B$2,400

WouldyouchooseAorB?

p Firstexperiment:choosebetweenAandB

$2,500A$2,400B$2,400

MostparticipantschooseB,thereforetheymustconsiderU(2,400)>0.33U(2,500)+0.66(2,400)+0.01U(0)

p Secondexperiment:choosebetweenCandD

$2,500$2,400CD

MostparticipantschooseC,thereforetheymustconsider0.33*U(2,500)+0.67*U(0)>0.34U(2,400)+0.66U(0)

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!

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

Objectionswiththetheoryofexpectedutility:the“framingeffect”

MoreontheframingeffectObjectsdescribedintermsofapositivelyvalencedproportionaregenerallyevaluatedmorefavorablythanobjectsdescribedintermsofthecorrespondingnegativelyvalencedproportion.Forexample,inonestudy,beefdescribedas“75%lean”wasgivenhigherratingsthanbeefdescribedas“25%fat”(LevinandGaeth1988)

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.

p OthersubjectsarepresentedwithoptionsCandD:n C:Ifthisprogramisadopted,400peoplewilldie.n D:Ifthisprogramisadopted,thereisaone-thirdprobabilitythatnobodywilldieandatwo-thirdsprobabilitythat600peoplewilldie.

TherobustexperimentalfindingisthatsubjectstendtopreferthesurethingwhengivenoptionsAandB,buttendtopreferthegamblewhengivenoptionsCandD.Note,however,thatoptionsAandCareequivalent,asareoptionsBandD.Subjectsthusappeartoberisk-averseforgainsandrisk-seekingforlosses,acentraltenetofprospecttheory.

Moneylotteriesandriskaversionp Conceptof“riskaversion”Supposeyoufacethefollowinglottery

10€AB4€

0€ certainty

IfyoupreferBtoA,thenyouare“riskaverse”,sinceforyou,u(4)>0.5*u(10)+0.5*u(0)

MoneylotteriesandriskaversionForanyoutcomeCi betweenbestandworst,therewillbesomeprobabilitypi,suchthat:

$1000(bestoutcome)Ci ~

$0(worstoutcome)

ForanyCi (e.g.$2,or$10)betweenbestandworstoutcome,therewillbesomeprobabilitypi,suchthatweareindifferentbetweenthecertaintyofCi andthegamblebetweenbestandworst.

Choiceunderuncertaintyp Expectedutilityprinciple:(u iscardinalandweworkwithuncertainty)

ABEu(A)≥Eu(B)

(Recallthatwithoutuncertaintyu wasordinalandwehadABu(A)≥u(B))

Supposethatanindividualfacesthefollowinglottery(gamble):x1

Whatistheexpectedvalueofthisgamble?x2

random

Choiceunderuncertaintyp ,whereisthemean

valueofthegamble(whatIwouldgainonaverage).p Supposethatyoualsohavethechoice

px1 +(1-p)x2

Thegambleandthesurebethaveexactlythesameexpectedvalue.Whichonewouldyouchoose?Theonethatgivesyougreaterutility.So,let’scomparetheutilitylevelsofthegambleandofthesurebet.

xxppxxE =-+= 21 )1()~( x

Choiceunderuncertaintyp Supposethattheutilityfromx1 isu(x1)andtheutilityfromx2

isu(x2).p Expectedutilityofthegamble:

p Utilityofthesurebet

)()1()()~( 21 xupxpuxEu -+=

))1(())~(()( 21 xppxuxEuxu -+==

Riskaversionp Wedefineanindividualasrisk-avertifheprefersgainingthe

expectedvaluewithcertaintythanincurringsomeriskbutgainingthesamevalueonaverage.

iffu(px1 +(1-p)x2)≥pu(x1)+(1-p)u(x2)

certainamount gamble

Riskaversionp Wedefineanindividualasrisk-avertifheprefersgainingthe

expectedvaluewithcertaintythanincurringsomeriskbutgainingthesamevalueonaverage.

iffu(px1 +(1-p)x2)≥pu(x1)+(1-p)u(x2)

Butthishappenswhenu(.)isaconcave function

Riskaversion

Decreasingmarginalutilityofincome.Theutilitygainfromanextraeuroislowerthantheutilitylossofhavingaeuroless.Henceriskaversion=thefearoflosing

Risk-lovingattitudep Risk-lovingattitude

iffu(px1 +(1-p)x2)≤pu(x1)+(1-p)u(x2)

Risk-lovingattitudep Risk-lovingattitude

iffu(px1 +(1-p)x2)≥pu(x1)+(1-p)u(x2)

Butthishappenswhenu(.)isaconvex function

Risk-lovingattitude

u u(x)

x1xx2Ex

Risk-loving

Riskneutralityp Wedefineanindividualasrisk-neutral

iffu(px1 +(1-p)x2)=pu(x1)+(1-p)u(x2)

Riskneutralityp Wedefineanindividualasrisk-neutral

iffu(px1 +(1-p)x2)=pu(x1)+(1-p)u(x2)

Thismeansthatu(.)isalinearfunction

Riskneutrality

Risk-neutral

Examplep SupposethatyoufaceachoicebetweenAandB,where

€10AB€4

IfyoupreferBtoA,canwesaythatyouarerisk-averse?

Examplep SupposethatyoufaceachoicebetweenAandB,where

€10AB€4

IfyoupreferBtoA,canwesaythatyouarerisk-averse?Answer:YES.Becauseforyouu(4)>0.5u(10)+0.5u(0)

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)

Certaintyequivalentp AgainconsiderachoicebetweenAandB.

Risk-averse€20B€10

AB€9€0B€8

Theexpectedvalueofthelotteryis10€.

Risk-averse€20B€10BA

AB€9€0B€8

AB€9BA€0B€8

AB€9BA€0B€8B~A

Certaintyequivalent (let’sdenoteity):itistheamountthatmakesmeindifferentbetweenthegambleandthesureamount.Itistheamountofmoneythat,ifgainedwithcertainty,providesthesameutilityasthegamble.

Riskpremiump AgainconsiderachoicebetweenAandB.

Riskpremium(π):thedifferencebetweentheexpectedvalueofthegambleandthecertaintyequivalent(:10€-8€=2€).

Riskpremium(π)p TheriskpremiumisthemoneyIabandoninordertohavemore

safety.

Or,inotherwords,thelossofincomethatcanbeconcededinordertogetridoftherisk(andobtainthecertaintyequivalent).

Itmeasuresthegapbetweentheexpectedvalueofthegambleandthecertaintyequivalent.Itis“positivelycorrelated”withriskaversion.

Insummary, and)~()()( xEuyuxu ==-π xy =+ π

Graphicalrepresentationu

Eu(x )amount you would sacrifice to eliminate the risk (π)

§ Expected payoff and the utility of expected payoff.

§ Expected utility and the certainty-equivalent

§ The risk premium

§ Utility values of two payoffs

Eu(x )

Graphicalrepresentation(similarfromMWG)

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)

Theprobabilitypremiumgraphically

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.

TheArrow-Prattmeasureofabsoluteriskaversion

Notethat:1.

2.rA (x)isafunctionofx,wherex canbethoughtofastheconsumer’scurrentlevelofwealth.Thuswecanadmitthesituationwheretheconsumerisriskaverse,riskloving,orriskneutralfordifferentlevelsofinitialwealth.

3.Wecanalsothinkabouthowthedecisionmaker’sriskaversionchangeswithherwealth.Howdoyouthinkthisshouldgo?Doyoubecomemoreorlesslikelytoacceptagamblethatoffers100withprobability½and−50withprobability½asyourwealthincreases?

Hopefully,youansweredmore.Thismeansthatyoubecomelessriskaverseaswealthincreases,andthisishowweusuallythinkofpeople,ashavingnon-increasingabsoluteriskaversion.

4.TheAPmeasureiscalledameasureofabsoluteriskaversionbecauseitsayshowyoufeelaboutlotteriesthataredefinedoverabsolutenumbersofdollars.Agamblethatofferstoincreaseordecreaseyourwealthbyacertainpercentageisarelativelottery,sinceitsprizesaredefinedrelativetoyourcurrentlevelofwealth.Wealsohaveameasureofrelativeriskaversion,

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

Application:Insurancep Exp.Cost=(1-π).(-αq)+π(-αq+α)=0

" q=π(thecosttotheconsumerof1euroofinsuranceisjusttheexpectedcostofprovidingthatcoverage)

Howmanyunitsofinsuranceshouldtheconsumerbuyiftheinsuranceisactuariallyfair?Findα tomaxexpectedutility.

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)

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