The Classical Paradigm and Behavioral Economics

INDIVIDUAL CHOICE:CHOICE UNDER RISKECN 111, Spring 2015Asen Ivanov1Risk vs. UncertaintyRisk (this lecture): DM doesnt know what will happen but there are objective probabilitiesroulette wheels, coin tosses, games of dicedoes not apply to most situations, but good starting pointUncertainty (later lectures): DM doesnt know what will happen and there need not be objective probabilitiesmost situations in life (investing, football outcomes, medical procedures, wars, marriage, etc.)2OBJECTS OF CHOICE3LotteriesLet z1,z2,,zn be outcomesFor simplicity, assume outcomes are monetary prizesE.g.:z1 is win 0z2 is win 5etc.Let p1,p2,,pn be nonnegative numbers that sum to 14LotteriesA lottery L=(z1,p1;z2,p2;;zn,pn) leads to:z1 with probability p1z2 with probability p2etc.

Objects of choice are lotteries.

5p1p2pnz1z2zn...........Expected Value of a LotteryDefinition:The expected value of a lottery L=(z1,p1;;zn,pn) is defined as follows: E(L)=z1p1+ z2p2 ++znpn.

6ExamplesL=(0,.4;10,.6)E(L)=0.4*0+0.6*10=6

L=(-5,.4;5,.5;10,.1),E(L)=0.4*(-5)+0.5*5+0.1*10=1.57assumptions on C8Key Assumptions on C(almost always made)WARP[Archimedean axiom (technical assumption)]Independence axiomInternal consistency assumptionProbably very appealing as normative assumptionControversial as descriptive assumptionYou dont need to know this assumption.You do need to know it is a key assumption in the theory of choice under risk.

9Conditions on C Related to DMs Attitude to RiskRisk aversionoften assumedRisk neutralityoften assumed because it simplifies the analysisRisk lovinglogically possible, but never really assumed (because empirically implausible)Constant Absolute Risk Aversionsometimes assumedConstant Relative Risk Aversionsometimes assumed

10Independence Axiom (optional)11What do you prefer: L1=($4000,0.8;$0,0.2) or L2=($3000,1)?L1L212

What do you prefer:L4=($4000,0.2;$0,0.8) or L5=($3000,0.25;$0,0.75)?L4L513

Independence AxiomWe will be a bit informalLet L1, L2, and L3 be three lotteries and q be a number between 0 and 1.Let L4 and L5 be lotteries obtained as follows:

Assumption (Independence Axiom):L1PL2 if and only if L4PL5

L1L3q1-qL4:L2L3q1-qL5:14Independence Axiom: Allais ParadoxExperiment (Kahneman and Tversky 1979):Subjects choose between:L1=($4000,0.8;$0,0.2) vs. L2=($3000,1)L4=($4000,0.2;$0,0.8) vs. L5=($3000,0.25;$0,0.75)80% of subjects choose L2 over L1.65% choose L4 over L5.Note that if q=0.25 and L3=($0,1), L4 and L5 can be obtained as in previous slide (verify this!)Thus, many subjects violate independence axiom.Allais paradox challenges descriptive accuracy of independence axiom15Attitude to Risk16What do you prefer: L=(0.5,0;0.5,10) or 5 for sure?L5 for sureIm indifferent17

Attitudes to RiskDefinition:DM is risk averse if, for any lottery L, DM prefers (E(L),1) to L, i.e., DM prefers getting E(L) for sure to taking her chances with L.

Definition:DM is risk neutral if, for any lottery L, DM is indifferent between (E(L),1) and L.

Definition:DM is risk loving if, for any lottery L, DM prefers L to (E(L),1).18ExampleConsider L=(0,0.5;10,0.5), which has expected value 5.

What is DMs preference between L and (5,1) if she is risk averse?How about if she is risk neutral?How about if she is risk loving?

19CARA (optional)Formal statement of CARA is a bit complicatedHowever, main idea is simple:DMs willingness to take risks involving adding/subtracting given amounts of money to/from her wealth is same at any level of wealthAn example should illustrate this idea

20Example (optional)Consider:Situation 1: DM is offered 50-50 chance to lose 5 or win 10 when her wealth is 100Situation 2: DM is offered 50-50 chance to lose 5 or win 10 when her wealth is 200According to CARA:DM takes the offer in situation 1 if and only if DM takes the offer in situation 2.I.e., DMs willingness to take the offer is the same regardless of her wealth.

21CRRA (optional)Formal statement of CRRA is also complicatedHowever, main idea is simple:DMs willingness to take risks involving adding/subtracting given proportions of her wealth to/from her wealth is same at any level of wealthAn example should illustrate this idea

22Example (optional)Consider:Situation 1: DM is offered 50-50 chance to lose 5% of her wealth or win 10% of her wealth when her wealth is 100Situation 2: DM is offered 50-50 chance to lose 5% of her wealth or win 10% of her wealth when her wealth is 200According to CRRA:DM takes the offer in situation 1 if and only if DM takes the offer in situation 2.I.e., DMs willingness to take the offer is the same regardless of her wealth.

23Comparing Risk Attitudes (optional)Definition:DM1 is more risk averse than DM2 if, for any lottery L and any amount of money x, the following holds:If DM1 weakly prefers L to (x,1), then so does DM2.

24Certainty Equivalent (optional)Definition (Certainty Equivalent):The certainty equivalent of a lottery L is the amount of money c(L), such that DM is indifferent between L and getting c(L) for sure.

25Example (optional)Consider L=(0.5,0;0.5,10).If DM:prefers 4.01 for sure to Lis indifferent between 4 for sure and Lprefers L to 3.99 for surethen c(L)=4.

26Would you take L1=(-10,0.5;11,0.5)?YesNo27

Would you take L2=(-100,0.5;,0.5)?YesNo28

Small vs. Large Stakes (optional)Assume WARP + independence axiom [+ Archimedean axiom].Then, even small degree of small-stakes risk aversion at a range of levels of wealth adds up to crazy large-stakes risk aversion (Rabin (2000)).This statement is a bit informal. E.g.:What do we mean by small degree of risk aversion or crazy risk aversion?What do we mean by adds up?An example should elucidate matters

29Example (optional)Consider again:L1=(-10,0.5;11,0.5)L2=(-100,0.5;,0.5)Assuming WARP + independence axiom [+ Archimedean axiom], if DM turns down L1 at all levels of wealth DM turns down L2 at all levels of wealth

30Small vs. Large Stakes: Implications (optional)If you:want to comply with WARP + independence axiom [+ Archimedean axiom] anddont want to be extremely risk averse for large-stakes gambles (e.g., you dont want to reject lotteries like L2 above),then you should be risk neutral for small-stakes lotteries [unless your small-stakes risk aversion occurs for some special reason only at your current wealth]

31Small vs. Large Stakes: Implications for Insurance (optional)Say, based on past statistics, probability is 0.01 that iPhone 5s (worth 400) is stolen from average iPhone ownerAssume DM is same as average iPhone 5s ownerDM can buy insurance for price P and eliminate riskAssume P=4.(Called actuarially fair price. In reality, P needs to be above this price for insurance company to make money.)DM has to choose between(-400,0.01;0,0.99) (no insurance)(-4,1) (insurance)Preferring to buy insurance requires risk aversion. (Why?)32Small vs. Large Stakes: Implications for Insurance (optional)Thus, insurance is probably bad idea for small-stakes risks (cell phone, laptop)Could be OK for large-stakes risks

33representation of c by a utility function34Representation By a Utility FunctionExpected Utility Theorem:C satisfies WARP and the independence axiom [+ the Archimedean axiom]if and only ifC can be represented by a utility function of the form: U((z1,p1;z2,p2;;zn,pn))=p1u(z1)+p2u(z2)++pnu(zn),where u:RR.

u is called DMs von Neumann-Morgenstern utility functionFor remaining propositions assume C satisfies WARP + independence axiom [+ Archimedean axiom] and let u be DMs von Neumann-Morgenstern utility function 35Non-Uniqueness of uProposition:Given any positive number, a, and any number b, v(z)=au(z)+b, can be used instead of u to represent C.

Example:If C can be represented byU((z1,p1;z2,p2;;zn,pn))=p1ln(z1)+p2ln(z2)++pnln(zn),it can also be represented by:U((z1,p1;z2,p2;;zn,pn))== p1[7ln(z1)-103]+p2[7ln(z2)-103]++pn[7ln(zn)-103]36Representation By a Utility Function and Risk AttitudeIf we make further assumptions about DMs risk attitude, we can be more specific about the shape of u

37Risk Attitude and Shape of uProposition:DM is risk averse/risk neutral/risk loving if and only if u is concave/linear/convex.

Pictures:

38CARAProposition:If C satisfies CARA, u(z)=-e-az, where a>0 is a parameter (e.g., a=2)

Example (using a=2):

If L1=(0,.5;1,.5), thenU(L1)=0.5(-e-2*0)+0.5(-e-2*1)-0.568.

If L2=(0,.7;2,.3), thenU(L2)=0.7(-e-2*0)+0.3(-e-2*2)-0.71.

Thus, DM prefers L1 over L2.39CRRAProposition:If C satisfies CRRA:either u(z)=z1-r, where r is a parameter different from 1 (e.g., r=0.7)or u(z)=ln(z)

Example (using u(z)=ln(z)):

If L1=(1,.5;10,.5), thenU(L1)=0.5ln(1)+0.5ln(10)1.151

If L2=(1,.7;20,.3), thenU(L2)=0.7ln(1)+0.3ln(20)0.9

Thus, DM prefers L1 over L2.40cognitive approach41Cognitive ApproachDM has pleasure meter or hedonimeter in her headIt assigns a level of desirability/happiness to each outcome z1, z2, etc.Von Neumann Morgenstern utility function u captures pleasure meter readings.I.e., u(z1), u(z2), etc. is happiness assigned to each outcome z1, z2, etc.DM computes expected happiness of each lotteryDM chooses lottery with highest expected happiness42Cognitive ApproachSimilar to case of consumption bundles, diminishing marginal utility of u makes sense:first 1 buys breadsecond 1 buys socksthird 1 buys stereoProvides justification for risk aversion

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