Exam 17: Choice and Markets in the Presence of Risk

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

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

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.

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)?

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

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.

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 $u ( x ) = x ^ { 0.75 }$ . 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?

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

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.

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

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?

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.

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

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.

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

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

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.

The risk premium is negative when tastes are risk averse.