Deck 17: Choice and Markets in the Presence of Risk

Question 1

Suppose that you face a gamble that has a payoff of $1000 with probability 0.2 and a payoff of $200 with probability 0.8.I approach you to sell you insurance with a premium of p and a benefit of

A)p=8, b=10
B)p=400, b=500
C)p=720, b=900
D)(a) and (b) are actuarily fair
E)(a) and (c) are actuarily fair
F)(b) and (c) are actuarily fair
G)All of the above.
H)None of the above.

Answer 1

All of the above.

Question 2

If the probability of the bad outcome is 0.5, the benefit level of actuarily fair insurance will be half the premium.

Answer 2

False

Question 3

Expected utility functions have to be concave if they are to represent risk averse tastes.

Answer 3

False

Question 4

Suppose an individual has state-independent tastes and invests in risky stocks rather than safe bonds.We can infer that he must be risk loving.

Question 5

Suppose that individuals with state-independent and risk-averse tastes insure each other through state-contingent trades.If there is no aggregate risk, the competitive equilibrium price will then result in actuarily fair insurance terms.

Question 6

Gamble A results in $10 with probability 0.4 and $30 with probability 0.6.Gamble B results in $20 with probability 1.If an individual prefers Gamble A to Gamble B, the independence axiom implies that he prefers Gamble C that gives $0 with probability 0.5, $10 with probability 0.2 and $30 with probability 0.3 to Gamble D that results in $20 with probability 0.5 and $0 with probability 0.5.

Question 7

Which of the following is true about an individual's choice of insurance assuming state-independent tastes?

A)Full insurance will be chosen from a full menu of actuarily fair insurance if tastes are risk averse.
B)No insurance will be chosen if the menu of insurance contracts is actuarily unfair and tastes are risk averse.
C)No insurance will be chosen if the menu of insurance contracts is actuarily unfair and tastes are risk-neutral.
D)(a) and (b) are true.
E)(a) and (c) are true.
F)(b) and (c) are true.
G)All of the above.
H)None of the above

Question 8

The certainty equivalent is less than the expected value of a gamble when tastes are risk averse.

Question 9

Which of the following is true about a risk-averse individual facing a full menu of actuarily fair insurance contracts to choose from?

A)The individual will "over-insure" if consumption is more meaningful in the good state.
B)The individual will "over-insure" if consumption is more meaningful in the bad state.
C)The individual will fully insure when tastes are state-independent.
D)(a) and (b) are true.
E)(a) and (c) are true.
F)(b) and (c) are true.
G)All of the above.
H)None of the above.

Question 10

The independence axiom implies that if I prefer a bottle of wine over a six pack of beer I will prefer half a bottle of wine with a bag of pretzels over half a sixpack of beer with a bag of pretzels.

Question 11

Risk averse individuals will fully insure to avoid risk.

Question 12

When tastes are risk averse, an individual will always choose less risk over more risk.

Question 13

When tastes are risk loving, a person will always choose a gamble that is riskier over one that is less risky.

Question 14

The risk premium is negative when tastes are risk averse.

Question 15

Suppose that individuals with state-independent and risk-averse tastes insure each other through state-contingent trades.The competitive equilibrium price will then result in actuarily fair insurance terms.

Question 16

Actuarily fair insurance reduces risk without changing the expected value of a gamble.

Question 17

The certainty equivalent of a gamble is negative when tastes are risk loving.

Question 18

Suppose an investor with state-independent tastes is offered the choice between investment A and investment B.Investment A offers profit of $2,000 with probability 0.4, $4,000 with probability 0.2 and $6,000 with probability 0.4.Investment B offers profit of $2,000 with probability of 0.5 and $6,000 with probability 0.5.If the investor is risk averse, he will choose investment A.

Question 19

Expected utility theory assumes that individuals have utility functions over a composite consumption good.

Question 20

Two related ways to quantify a person's degree of risk aversion are the certainty equivalent and the risk aversion premium.

Question 21

Which of the following can explain the choice to gamble in casinos?

A)Gamblers have risk loving tastes.
B)Gamblers have risk-averse but state-dependent tastes.
C)The casino is operating at a loss.
D)(a) and (b)
E)(a) and C
F)(b) and (c)
G)All of the above.
H)None of the above.

Question 22

Suppose you rent an apartment and are worried about a break-in that results in theft of your property.Suppose your monthly consumption level is currently $4,000 but a break-in would result in you having to finance your purchase of replacement property -- and this would reduce your current consumption to $2,000 per month.There is a 10% chance of a break-in, and your tastes can be modeled with the expected utility form using the function .
a.What is the utility of the expected value of the gamble you face, and what is the expected utility of the gamble?
b.How does your answer to (a) change if the probability of a break-in increases to 20%?
c.What is the certainty equivalent and the risk premium in each case?
d.What equation would you have to solve to get the answer to the following: How much would you be willing to pay to keep the crime rate in your area from increasing (i.e.to keep the probability of a break in to 10% rather than have it rise to 20%) assuming there is no rental insurance available in your area?
e.What would you be willing to pay to avoid the increase in the crime rate if there is a full menu of actuarily fair rental insurance available at all times?

Question 23

Suppose, after undergoing genetic testing, you discover that you have a health condition that could result in the emergence of a disability which would make it impossible for you to continue to work.The probability of this happening is 50%.Currently your expected lifetime earnings are $5,000,000, but if the disability hits, your expected lifetime earnings will consist primarily of income earned from government support programs -- and will not add up to more than $1 million.
a.Suppose that you are risk averse and your tastes are state-independent.Illustrate your expected utility in a graph with lifetime consumption on the horizontal and utility on the vertical axis.
b.Illustrate how much you would be willing to pay for full insurance.
c.Illustrate what you showed in (b) in a different graph that has consumption in the "good" state on the horizontal and consumption in the "bad" state on the vertical.
d.What would a full menu of actuarily fair insurance contracts look like in your graph from part (c)? Where would you optimize in that graph?
e.Now suppose that you believe consumption will be more meaningful if the health condition does not materialize.What changes in your graph from part (d)?

Question 24

Suppose you rent an apartment and are worried about a break-in that results in theft of your property.Suppose your monthly consumption level is currently $4,000 but a break-in would result in you having to finance your purchase of replacement property -- and this would reduce your current consumption to $2,000 per month.There is a 10% chance of a break-in.
a.On a graph with "consumption" on the horizontal and "utility" on the vertical axis, illustrate a utility/consumption relationship that is consistent with risk averse tastes.
b.On your graph, illustrate the utility in the "good" state, the utility in the "bad" state and the expected utility of facing the gamble.
c.Which of these changes when the probability of a break-in increases to 20%?
d.A renter's insurance policy consists of a premium p and a benefit level b.What is (b,p) for full, actuarily fair insurance before and after the increase in risk?
e.True or False: You are more likely to buy actuarily fair full insurance after the increase in risk than before.

Question 25

Suppose you and I are the only two individuals in the world and we both face individual risk in the following way: I get more consumption in state 1 than in state 2 while you get more consumption in state 2 than in state 1.
a.Suppose we are both risk averse and our tastes are state-independent.Will we fully insure one another in a competitive equilibrium?
b.How does your answer change if there is aggregate risk in the sense that overall consumption is higher in state 2 than in state 1?
c.Is it possible that we insure each other if our tastes are risk neutral and state-independent? If so, are the terms actuarily fair?
d.Suppose that there is no aggregate risk but our tastes are state-dependent.How might we fully insure each other if our beliefs about the probability of each state differ?

Question 26

Suppose, after undergoing genetic testing, you discover that you have a health condition that could result in the emergence of a disability which would make it impossible for you to continue to work.The probability of this happening is 50%.Currently your expected lifetime earnings are $5,000,000, but if the disability hits, your expected lifetime earnings will consist primarily of income earned from government support programs -- and will not add up to more than $1 million.
a.Suppose your tastes are state-independent and the function can be used to represent your tastes in the expected utility form.Are you risk averse?
b.What is the highest premium you would pay to get fully insured?
c.What is the equation (in terms of -- consumption in the bad state -- and -- consumption in the good state) that defines the full menu of actuarily fair insurance contracts?
d.Set up the optimization problem that you would solve as you choose among actuarily fair insurance contracts.
e.Solve the optimization problem.What does this imply will be the insurance contract (b,p) that you buy -- where b is the benefit level and p is the insurance premium?
f.Finally, suppose you had state dependent tastes and that the functions and allowed us to use the expected utility form to represent your tastes.How does your answer to (e) change?