behavioral economics - topic2a - expected utility

8/11/2019 Behavioral Economics - Topic2a - Expected Utility

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Topic 2A: The Standard Model: Expected Utility

Econ 4660: Behavioral EconomicsProfessor Ted ODonoghue

Spring 2014

() Expected Utility Spring 2014 1 / 28
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Topic 2: Choice Under Uncertainty

Topic 2A: The Standard Model: Expected Utility

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Dening the Environment

Denition: A lottery(orgambleor risky prospect) is a set of possible

outcomes and a probability of each outcome occurring.

Example A: If you go to a roulette table and place $10 on BLACK, youhave just purchased a lottery. With probability 18/38 you will receive $20,

and with probability 20/38 you will receive $0.

Example B: When you buy a new car, you are buying a lottery. Forinstance, it might be that with probability 1/2 it will be a car that youlove (high value), with probability 2/5 it will be a car that only adequately

serves your needs (low value), and with probability 1/10 it will be alemon (worthless).

Example C: More abstractly, you might get x1 with probability p1, x2 withprobability p2, and x3 with probability p3 (where p1+p2+p3 =1).

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Ways to write lotteries:

As a probability tree.

As a vector of outcome-probability pairs:

Example A: Payo= ( $20, 18/38 ; $0, 20/38 ).

Example B: Car Value = ( high, 1/2 ; low, 2/5 ; worthless, 1/10).

As a type of equation:

Example A:

Payo = $20 with prob 18/38$0 with prob 20/38

Example B:

Car Value =

8


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In the realm of choice under uncertainty, lotteries are the objects of choice.

Next step: Develop models of behavior.

Suppose you face a choice between two lotteries. How do you decide?

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Model #1: Expected Value Theory (EV)

A possible model of behavior: People choose the option with the largestexpected value.

Denition: The expected valueof a lottery x= ( x1, p1 ; ... ; xn , pn ) is

EV(x) n

i=1

pi xi.

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A Problem for EV

St. Petersburg ParadoxConsider the following bet: Im going to ip a coin, and Im going to keep

on ipping it until I ip a HEADS. Then youll be paid as a function ofhow many times we ip. Specically:

If I immediately ip a HEADS, you get $2.

If I ip one TAILS and then a HEADS, you get $4.

If I ip two TAILS and then a HEADS, you get $8.

If I ip three TAILS and then a HEADS, you get $16.

And so forth....

How much are you willing to pay for this bet?^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^The paradox:

The EVof this bet is , but people are unwilling to pay much for it.Hence, EV is not a good description of peoples choices.

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Whats Missing from EV? Risk Aversion

Denition: A person is risk-averseif for any (risky) lottery x she prefersto have EV(x) for sure instead of lottery x.

Denition: A person is risk-neutralif for any (risky) lottery x she is

indierent between EV(x) for sure vs. lottery x.

Denition: A person is risk-loving if for any (risky) lottery xshe prefers tohave lottery x instead ofEV(x) for sure.

=) Evidence suggests people are risk-averse.

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Model #2: Expected Utility Theory (EU)

A second model of behavior: People choose the option with the largestexpected utility.

Denition: The expected utilityof a lottery x= (x1, p1; ...; xn , pn ) is

EU(x) n

i=1

pi u(xi).

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Basic Implications of EU

Could an expected-utility maximizer (EU maximizer) choose(100, 12 ; 0,

12 ) over (200,

12 ; 0,

12 )?

.

Point: If we put no restrictions on the utility function u(x), thenexpected-utility theory can explain any individual choice.

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Three Comments:

(1) We say that expected-utility theory can explain a choice if thereexists a utility function u(x) such that an EU maximizer with utilityfunction u(x) would make that choice.

(2) Even when we put no restrictions on the utility function u(x),although an individual choice cannot violate EU, combinations of choicescan violate EU.

E.g., (100, 12 ; 0,12 ) (200,

12 ; 0,

12 )

AND (100, 13 ; 500, 23 ) (200, 13 ; 500, 23 )

(3) When we put restrictions on the utility function u(x), then even anindividual choice could violate EU.

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A natural restriction: We virtually always assume that more is better

that is, uis increasing in x.Denition: Lottery x dominates lottery y (or equivalently, lottery y isdominated by lottery x) if for any amount zthe probability of gettingat least z in lottery x is larger than the probability of getting at least z inlottery y.

Examples:

(200, 12 ; 0,12 ) vs. (100,

12 ; 0,

12 )

(200, 1

2

; 0, 1

2

) vs. (150, 1

2

; 50, 1

2

)

(200, 13 ; 150,13 , 75,

13 ) vs. (150,

12 ; 75,

12 )

Result: Under expected utility with more is better, a person will neverchoose a dominated lottery.

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To incorporate risk-aversion (or risk-seeking), we must assume more about

the utility function.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^Result: Under expected-utility theory, a person is risk-averse if and only ifher utility function is concave (u00(x) < 0).

Result: Under expected-utility theory, a person is risk-neutral if and only ifher utility function is linear (u00 (x) =0).

Result: Under expected-utility theory, a person is risk-loving if and only ifher utility function is convex (u00(x) > 0).

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^Note: In this class, whenever we say that someone is a risk-averse (orrisk-neutral or risk-loving) expected-utility maximizer, well implicitlyassume more is better as well.

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5 Features/Issues of Expected Utility

(1) The utility function is unique up to a positive ane transformation.

Denition: The function u(x) is a positive ane transformation of thefunction u(x) ifu(x) =a u(x) +bfor some a > 0 and any b.

Result: Ifu(x) is a positive ane transformation ofu(x), then theyreect the same preferences that is, for any pair of lotteries x and y,

EU(x) > EU(y) if and only ifEU(x) > EU(y).

(2) Can u(x) be negative? Can EU(x) be negative?

Answer: sure.

Point: The level of utility doesnt matter, only the comparisons.

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(3) Integration: EUoperates on nal wealth states.

Proper way to use expected-utility theory:

Let wbe a persons wealth.

Let x=(x1, p1; ...; xn , pn ) be a lottery (or gamble or risky prospect).Note: x yields income xi with probability pi.

EU theory says evaluate prospect x according to:

EU(x; w) n

i=1

pi u(w+xi).


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(4) Closely related: EUoperates on reduced lotteries.

Example: If you face the possibility of both lotteryx= (100, 1/2; 100, 1/2) and y= (90, 1/2; 90, 1/2), you must rstcombine these into a reduced lottery before applying EU.

If they are independent, the reduced lottery is

(190, 1/4; 10, 1/4; 10, 1/4; 190, 1/4).

If they have perfect positive correlation, the reduced lottery is

(190, 1/2; 190, 1/2).

If they have perfect negative correlation, the reduced lottery is

(10, 1/2; 10, 1/2).


F /I f E d U ili
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(5) We can measure the degree of risk aversion.

Goal: Wed like to make statements of the form, If person A is morerisk-averse than person B, then Ais less likely to accept a particulargamble.

Question: How to formalize more risk averse?

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^Denition: The coecient of risk aversion at wealth w is

r(w) = u00 (w)

u0(w) .

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^SupposewA is person As wealth and wB is person Bs wealth. Ifr(wA) > r(wB) then person A is less likely than person Bto accept arisky gamble.


5 F /I f E d U ili
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Note: A persons risk aversion depends on both her utility function andher wealth.

r(w) = u00 (w)

u0(w) .

How is ones willingness to accept a gamble likely to depend on my wealth?

Economists often assume risk aversion decreases with wealth:

" w=)# r(w).


A I F i l F
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An Important Functional Form

Denition: The CRRA utility function is

u(z) =(z)1

1 for 2 [0,).

Some features:

For =0, equivalent to u(x) =x, or risk neutrality.

For =1, equivalent to u(x) =ln x.

Note that r(w) = /w, hence " =)" r(w) and " w =) # r(w).

Person feels the same about the following gambles:

gamble ($11, 1/2; $10, 1/2) from wealth $1, 000



Empirical studies suggest in the 1-5 range.() Expected Utility Spring 2014 19 / 28

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Application: Demand for Insurance

Suppose you have wealth $1000.

But there is a 10% chance that you will suer a loss of $250.Hence, you face the lottery ( 1000, .9 ; 750, .1 ).

Insurance agent: I will fully insure you if you pay me p.

Implication: If buy insurance, you face the lottery ( 1000 p , 1 ).

What is your willingness to pay for full insurance?

Answer: Its the p such that

( 1000 p, 1 ) ( 1000, .9 ; 750, .1 )

oru(1000 p) = (.9) u(1000) + (.1) u(750).


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Suppose instead:To insure against the entire $250 loss, it costs p.

To insure against proportion of the $250 loss, it costs p.

What is the optimal ?

Because is continuous, use calculus:

Step 1: Derive nal lottery as a function of.

Step 2: Calculate expected utility as a function of.

Step 3: Use calculus to solve for the optimal .


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As a function of the you choose, nal lottery is:

x() = ( 1000 p , .9 ; 750+ (250 p) , .1 ).

As a function of the you choose, expected utility is:

EU(x()) = (.9) u(1000 p) + (.1) u(750+ (250 p)).


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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^Denition: Insurance is actuarially fairif the premium equals the expectedpayout.

(Insurance is actuarially unfairif the premium is larger than the expected

payout, and insurance is actuarially overfairif the premium is smaller thanthe expected payout.)^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Lets assume insurance is actuarially fair.

Here, actuarially fair insurance means p= (.10)(250) = $25.


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Substituting p= $25 into EU(x()):

EU(x()) = (.9)u(1000 25) + (.1)u(750+225).

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Conclusion: If insurance is actuarially fair, and if you are risk averse, thenyou will choose to fully insure.


Application: Demand for Financial Assets
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Suppose you have wealth 1000, and that you plan to invest it for one year

and then consume it. Two assets in which to invest:A risk-free assetthat yields a certain return of 0%.

A risky assetthat yields a return of 50% with probability 1/2 and areturn of40% with probability 1/2.

Let be the amount you invest in the risky asset, so 1000 is theamount you invest in the risk-free asset. (Assume 0 1000.)

If do well, nal wealth =

(1000 )(1.00) +(1.50) =1000+(.50).

If do poorly, nal wealth =(1000 )(1.00) +(0.60) =1000 (.40).


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As a function of the you choose, you face lottery:

x() = ( 1000+(.50) , 1

2 ; 1000 (.40) ,

1

2 ).

As a function of the you choose, your expected utility is:

EU(x()) = 1

2 u(1000+(.50)) +

1

2 u(1000 (.40)).


Expected Utility and Revealed Preferences
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Expected Utility and Revealed Preferences

Recall that a theoretical goal is to prove theorems of the form: A personsbehavior can be described by a model if and only if the persons choices

satisfy some set of axioms.

Axioms (properties about a persons choices over lotteries):

(1) Preferences are complete and transitive.(2) Preferences are continuous.

(3) Independence axiom: Ifx % y, then for any lottery z,(1 )x+z % (1 )y+zfor any 0 < < 1.

Theorem: A persons behavior can be described by expected utility theoryif and only if her choices over lotteries are complete, transitive, continuous,and satisfy the independence axiom.


Expected Utility A Final Thought
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Expected Utility A Final Thought

Many people view EUto be a good normative theory.

EUseems to fail as a positive (descriptive) theory. Thats where well gonext....

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behavioral economics - topic2a - expected utility

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