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Lecture 5Consumer Choice under uncertainty

- Simple lottery
- A simple lottery L is a list L(p1pn) with pn?0

for all n and , where pn is the probability of

the outcome n occurring - We can define more complex lotteries (lotteries

over lotteries) - Compound lottery
- Given k simple lottery , k1K and some

probability that that a lottery Lk occurs, then

we can define a compound lottery , which is the

risky alternative that give Lk with probability

Lotteries

- Example
- Prize100
- Compound lottery A
- L1(100,0,0) occurs with probability 1/3
- L2(1001/4, 100 3/8, 1003/8) with probability

1/3 - L3((1001/4, 1003/8, 1003/8) with probability

1/3 - Compound lottery B
- L1(1001/2, 1001/2, 0) with probability ½
- L2(1001/2, 0, 1001/2) with probability ½
- Do you prefer A or B?

Reduced Lotteries

Reduced lotteries

- Given all the possible outcomes, we can apply the

same logic to compute the probability of each

outcome and hence find the Reduced Lottery

Reduced lotteries

Preference over compound lotteries

- Since the consumer only cares about the

distribution of final outcomes, he will be

indifferent between to Compound Lotteries that

deliver the same Reduced Lottery - Do you agree?

Preferences over lotteries

- Remember when we defined preferences over a set

of goods. - Again now we have to define preferences but over

a space of lotteries - AXIOMS - We require
- Continuity - given two lotteries, small changes

in probabilities do not change the ordering

between two lotteries - Indipendence if we mix each of the two lotteries

with a third one, the preference ordering of the

two resulting mixtures does not depend on the

particular third lottery that is used. - Note the difference with the consumer choice

under certainty here when we consider three

alternative lotteries L1, L2 and L3, the consumer

cannot consumer L1 and L2 together or L1 and L3

together etc. but he has to choose between

mutually exclusive alternatives!

Expected Utility

Expected Utility

- Expected Utility theorem
- If preferences over lotteries satisfy the

continuity and the independence axioms, then

preferences can be represented by a utility

function with expected utility form - Indifference curves
- Since the Von-Neuman-Morgestern utility is linear

in probabilities, if we represent the

indifference curves on the lottery space, we will

obtain parallel straight lines - If not straight lines indipendence axiom not

satisfied - If not parallel indipendence axiom not satisfied

Consumer choice under uncertainty

- If preferences admit expected utility form

representation, then the consumer choice under

uncertainty reduces to the maximisation of his

expected utility, given his budget constraint

Money Lottery and risk adversion

- Solving problems under uncertainty we can see

that individuals show some degree of risk

aversion - Where does risk aversion come from?
- In other words, which property of the utility

function in the expected utility framework

implies risk aversion? - Suppose that we define utility over monetary

lotteries. Let x be certain amount of money an

individual receives - And u(x) the utility associated x. We can show

that risk aversion is implies by CONCAVITY of

u(x).UNIQUENESS - Is the utility function unique?
- NO - - ANY MONOTONIC TRANSFORMATION REPRESENTS

THE SAME PREFERENCES - UTILITY IS AN ORDINAL CONCEPT NOT A CARDINAL

CONCEPT!

Utility function

Risk aversion

- Concave utility function

Utility function

Risk Aversion

- Risk aversion can also be seen using two other

concepts - Certainty equivalent the sure amount of money

that you are willing to accept instead of the

lottery - Probability premium the excess in probability

over fair odds that makes an individual

indifferent between a certain outcome and a

gamble - If an individual is risk averse you expect that
- The sure amount of money he is willing to accept

instead of the gamble should be less then the

expected value of the money he would get from the

gamble (he is willing to take less and avoid the

risk) - An individual accepts the risk only if he is

offered better than fair odds

Measures of Risk Aversion

Measures of Risk Aversion

Risky choices

Risky Choices

- Examples of risky choices
- Insurance
- Investment in risky asset
- Comparisons of different payoffs
- Suppose that you are faced with the choice of

comparing different risky investments - Which type of statistics about the lottery would

you use to choose among investments?

Risky Choices

- level of return (average)
- dispersion of returns
- If you know the distribution function, you can

use this information to compare lotteries

Stochastic dominance

- First order stochastic dominance F(.)

first-order stochastically dominates G(.) if

Stochastic dominance

- Now, suppose that you can change F(.) in a way

that preserves the mean but changes the variance.

Suppose that G(.) is the function that you obtain

from this transformation, that is G(.) is a

mean-preserving spread of F(.) - Ex F(.) such that with probability p1/2 you get

2 and with prob p1/2 you get 3, hence the

average payoff is 5/2 - Let be G(.) such that with probability p1/4 you

can get (1,2,3,4) hence the average payoff is

again 5/2 - Would you prefer F(.) or G(.)?

Stochastic dominance

- If you are risk averse, you will prefer F(.) to

G(.). - Indeed, we can prove that F(.) second-order

stochastically dominates G(.) - Definition
- Second order stochastic dominance given F(.) and

G(.) with the same mean, F(.) second-order

stochastically dominates (or is less risky than)

G(.) if

Stochastic dominance

- Result if G(.) is a mean preserving spread of

F(.), then F(.) second-order stochastically

dominates G(.) - Proof
- Let x be a lottery distributed according to F(.).

Suppose that we further randomize x so that the

final payoff is xz where z is distributed

according the the function H(z) with zero mean.

Therefore, xz has the same mean as x but

different variance. We define G(.) final reduced

lottery, i.e. the function that is assigning a

probability to each x using the transformation of

F(.) we have just described. Hence G(.) is a mean

preserving spread of F(.).

Stochastic dominance

Expected Utility theory a calibration exercise

- Consider the following gamble
- You can loose 10 with probability 50 and gain

11 with probability 50 - Do you accept this gamble?
- Consider now these alternative gambles
- Loose 100 with 50 prob. and win 110 with 50

prob. - Loose 100 with probability 50 and win 221 with

50 prob. - Loose 100 with probability 50 and win 2000

with 50 prob. - Loose 100 with probability 50 and win 20,000

with 50 prob. - Loose 100 with probability 50 and win 1

million with 50 prob. - Loose 100 with probability 50 and win 2

millions with 50 prob. - Which bet will you be willing to accept?

Source M. Rabin and R.H.Thaler (2001),

Anomalies- Risk Aversion, Journal of Economic

Perspectives, pages 219-232

Expected Utility theory a calibration exercise

- Problem the expected utility theory delivers

implausible excessive degree of risk aversion!

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