Question

Suppose Hannah is strictly risk averse with a utility function u over monetary amounts (y):

u(y)=y​^(1/2)

Hannah is facing a risky situation: Either nothing happens to her wealth of $576 with probability 3/4 or she losses everything (so ends up with $0) with probability 1/4.

Question 1

What is the expected payoff that Hannah is facing? Provide the numerical value.

Numeric Answer:

Question 2

What is Hannah's expected utility in this gamble? Provide the numerical value.

Numeric Answer:

Question 3

What is Hannah's certainty equivalent for the gamble? Provide the numerical value.

Numeric Answer:

Question 4

Suppose Hannah can purchase insurance for 25 cents per dollar of coverage to insure herself against the bad outcome in the gamble. Check all correct statements.

Multiple answers:You can select more than one option

A

25 cents per dollar of coverage is not the actuarially fair premium.

B

The probability of the bad state is the same as the insurance premium per dollar of coverage.

C

Hannah would never buy insurance at such a premium.

D

The insurance premium of 25 cents per dollar of coverage is actuarially fair.

Question 5

Suppose Hannah purchases full coverage (i.e. $576 worth of coverage) from the insurance company. Check all correct statements.

Multiple answers:You can select more than one option

A

With full coverage Hannah faces the same amount of money in both states of the world.

B

With full coverage Hannah would be worse off than without insurance.

C

Hannah would pay the insurance company $144.

D

Hannah would pay the insurance company $576.

E

Hannah would pay the insurance company $432.

Question 6

What is the optimal amount of coverage that Hannah will purchase at a premium of 25 cents per dollar of coverage? Provide the numerical value.

Numeric Answer:

Question 7

What is Hannah's expected utility if she purchases full coverage at 25 cents per dollar of coverage? Use two decimals in your numerical answer.

Numeric Answer:

Question 8

Suppose the insurance premium per dollar of coverage is no longer 25 cents, but 50 cents. Check all the correct statements.

Multiple answers:You can select more than one option

A

Hannah would still purchase full coverage.

B

Hannah would purchase less than full coverage.

C

Hannah is better off choosing a coverage of $500 than full coverage.

D

Hannah is better off purchasing no insurance than full coverage.

E

Hannah is better off choosing a coverage of $500 than $300.

F

Hannah is better off choosing a coverage of $300 than no insurance.

G

Hannah is better off choosing a coverage of $500 than no insurance.

Question 9

Suppose we depict Hannah's insurance problem in the state-contingent space with the payoffs in the good state of the world on the horizontal axis. Check all the statements that are true.

Multiple answers:You can select more than one option

A

Hannah's indifference curves in the state-contingent space have a slope with an absolute value of 3 at the 45 degree line.

B

With an insurance premium of 50 cents per dollar of coverage Hannah's budget constraint has a slope with an absolute value of 1/2.

C

With an insurance premium of 50 cents per dollar of coverage Hannah's budget constraint has a slope with an absolute value of 1.

D

With an insurance premium of 25 cents per dollar of coverage Hannah's budget constraint has a slope with an absolute value of 1/3.

E

With an insurance premium of 25 cents per dollar of coverage Hannah's budget constraint has a slope with an absolute value of 3.

F

Hannah's indifference curve is always tangent to her budget constraint.

Answer 1

Question 1.

Calculation of expected payoff = (Probability of nothing happen * value of wealth) + (Probability of Loss happening * Value of loss)

(3/4 * $576 + 1/4 * 0) = $432

Hence expected payoff is $432.

Question 2.

Calculation of expected utility = P (w) * U (w) + P(l) * U (l)

Where: P(w) = Probability of Nothing happen

U (w) = Utility of Nothing happen

P (I) = Probability of Loss

U(l) = Utility of loss

Utility function = U(y) ^(1/2)

[3/4 * (576)^1/2 + 1/4 * (0)^1/2]

= $18

Hence expected utility is = $18