Consumer Choice With Uncertainty Part II: Expected Utility & Jensen’s Inequality

Agenda:

1. From Expected Value to Expected Utility: The VNM

2. Jensen’s Inequality: Risk Aversion & Our willingness to pay to reduce risk!

You make an initial investment of $10,000

A: The market goes up, gain $1,000 = 20%B: The market stays the same, gain $0 = 50%C: The market goes down, loose $5,000 = 30%

What is the EXPECTED VALUE of your position?

($11,000 x .2) + ($10,000 x .5) + ($5,000 x .3) = $8,700

1

N

i ii

E X X p

Expected Value to Expected Utility: Investment Example Revisited

U MWhat is the EXPECTED UTILITY of your position?

.2 11,000 .5 10,000 .3 5,000 92.19

What units?

Key Tool: Decision TreesNodes

Branches

.2

$11,000

.5

$10,000

.3

$5,000

($11,000 x .2) + ($10,000 x .5) + ($5,000 x .3) = $8,700

1

N

i ii

E X X p

=1

11,000 10,000 5,000

.5 10,00 .3 5.2 11,00 ,000 90 .190 2

What is the EXPECTED UTILITY of your position?

What is the EXPECTED VALUE of your position?

Von Neumann-Morgenstern Expected Utility Model

Expected utility is the weighted average of the utilities of the different possible outcomes.Message: We make decisions based on utilities!

1 2( ) ( ) (1 ) ( ); 1E U M pu M p u M p For two outcomes:

Expected utility is probability of outcome M1 times the utility of the outcome M1 plus 1- the probability of outcome 1 times the utility of the outcome M2. The probability of outcome 1 must be less than or equal to 1.

1 1

( ) ( ); 1N N

i i ii i

E U M p u M p

General form for N different possible outcomes:

Expected utility is the sum from 1 to N of the probability of the outcome times the utility of the outcome. The sum of all of the probabilities must equal 1 (collectively exhaustive).

You make an initial investment of $10,000

A: The market goes up, gain $1,000 = 20%B: The market stays the same, gain $0 = 50%C: The market goes down, loose $5,000 = 30%

What is the EXPECTED VALUE of your position?

($11,000 x .2) + ($10,000 x .5) + ($5,000 x .3) = $8,700

1

N

i ii

E X X p

Expected Value to Expected Utility: Investment Example Revisited

U M

What is the EXPECTED UTILITY of your position?

.2 11,000 .5 10,000 .3 5,000 92.19

What is the UTILITY of the EXPECTED VALUE?

8,700 93.27U

Jensen’s Inequality

Johan Ludwig William Valdemar Jensen

E u M u E M For any concave function, the expected value of the

function is less than the function of the expected value.

E u M

u E M “The key insight of the theory is that the expected values of the outcomes of a set of alternatives need not have the same ranking as the expected utilities of the alternatives.” Frank p. 181

You start with $40 and have a 50% chance of getting or losing $30 (Frank p. 182)

.5(18) .5(38) 28

What is your expected wealth?

.5(40 30) .5(40 30) 40

What is your expected utility?

Should you take the gamble?

(40) 32 28u No!

Risk averse people don’t take fair gambles

At what odds would you take the gamble?

(40 30) (1 )(40 30) 52

(10) (1 )(70) 52

(10 70) 70 52

( 60) 18

18 / 60 .3

p p

p p

p

p

p

.7 - .5 = .2 = probability premium: the excess over fair odds to get you to gamble.

chord

What is the UTILITY of your expected wealth? (read the answer off the graph)

(40) 32u

Because we are risk averse we are willing to pay MORE than the expected loss to reduce risk!

→ gains from trade!!

The Different Shapes of Risk

Risk Averse

Risk Neutral Risk Seeking

Test yourself:Does Jensen’s Inequality hold for all types of risk preferences?