Economics 201 Microeconomics

Tutorial 4

Question 1:

A firm with utility of wealth function U(w) = Sq Rt (W) = W^0.5 owns an airplane valued at $64m and other risk- free assets valued at $36m (i.e. the firm’s initial wealth is $100m). Assume the probability the plane crashes during the next year is 0.25. If this occurs the value of the plane is zero.

(a) What is the firm’s expected wealth?  

(b) What is the firm’s expected utility of wealth?

(c) What is the minimum premium an insurance company will be willing to charge the company for replacing the plane?

(d) What is the maximum premium the firm would be willing to pay to fully insure the plane, i.e. to get a replacement plane in the event of a crash.

Question 2:

Suppose Wendy’s current wealth (W) is $8,000 and she obeys the principles of expected utility theory. Her utility function is defined as U(W) = 2*W. Suppose she is offered a gamble where she can win $4000 with one-half chance or lose $2000 with one half-chance. Wendy:

(a) Will not accept this gamble since she is risk averse.

(b) Will not accept this gamble because the expected value of the gamble is negative $1000.

(c) Will accept this gamble because the expected value of this gamble is positive $1000.

(d) Will not accept the gamble because the utility she gets from holding on to her $8000 exceeds what she gets from accepting the gamble.

Question 3:

Assume, Elizabeth’s utility function is: U(W) = W^0.5 (square root of W) and she makes decision on the basis of expected utility theory. She is considering two job proposals:

Alternative 1: take a job at a bank with a certain salary of $45,000 per annum.

Alternative 2: take a job with a start-up company, get a base salary of $4,000 per annum a plus a bonus of $100,000 per year with probability 0.5 (otherwise bonus = $0). This implies with 0.5 chance she gets $4000 and with 0.5 chance she gets $104000.

Show that Elizabeth will prefer Alternative 1 with the sure payment over Alternative 2 with the probabilistic payment. What is the minimum sure payment Elizabeth is willing to accept in order to forego Alternative 2? What happens if we change the probabilities associated with Job 2 in the following way. With 0.4 chance Elizabeth gets $4000 and with 0.6 chance she gets a bonus and receives $104,000?

Question 4:

Dorian currently has $10,000. He is considering a lottery where he might win $4000 with 0.5 chance or he might lose $4000 with 0.5 chance. Dorian’s utility function is given by U(W) = W^0.5 (square root of W). What is Dorian’s certainty equivalent and what is the maximum risk premium he should be willing to pay?

Question 5:

Jim Holtzhauer is playing the tables at Vegas. He currently has $4000, which I will denote as 4K in order to keep things simple. He is looking at a bet where with ½ chance he can win 2K while with ½ chance he will lose 2K.  Jim’s utility of wealth function is given by U(W) = (W)². (U(W) is equal to the W Squared). Based on this information, you would conclude that:

Jim would take the bet since his expected utility from taking the bet is 20K² utils while the expected utility of staying with 4K is 16K² utils.

Jim would not take the bet since his expected utility from taking the bet is 16K² utils while the expected utility of staying with 4K is 20K² utils.

Jim would not take this bet since he is risk averse and his expected utility from taking the bet is negative.

Jim would take this bet because the expected value of the gamble: ½ chance of winning 2K and ½ chance of losing 2K is positive.

Question 6:

Consider a gamble that pays $40 with 0.5 chance and $100 with 0.5 chance. Suppose Oliver is risk loving and has utility function given by U(W) = W².  Heidi however is risk neutral and has utility function given by U(W) = W. Tingmeng is risk averse. Her utility function is U(W) = Square Root (W). Suppose that each of the three have $70 at their disposal. What is the certainty equivalent of each? What is the risk premium for each?