Jack is a talented investor, but his earnings vary considerably from year

to year. In the coming year he expects to earn either $250,000 with good

luck or $90,000 with bad luck. Somewhat oddly, given his chosen profession,

Jack is risk averse, so that his utility is equal to the square root of his

income. The probability of Jack's having good luck is 0.5.

a)What is Jack's expected utility for the coming year?

(b)What amount of certain income would yield the same level of utility

for Jack as the expected utility in part(a)?

Jack meets Janet, whose situation is identical in every respect. She's

an investor who will earn $250,000 in the next year with good luck and

$90,000 with bad, she's risk averse with square-root utility, and her probability

of having good luck is 0.5. Crucially, it turns out that Jack and Janet

invest in such a way that their luck is completely independent. They

agree to the following deal. Regardless of their respective luck, they will

always pool their earnings and then split them equally.

(c)What are the four possible luck-outcome pairs, and what is the

probability of reaching each one?

(d)What is the expected utility for Jack or Janet under this arrangement?

(e)What amount of certain income would yield the same level of utility

for Jack and Janet as in part(d)?

Incredibly, Jack and Janet then meet Chrissy, who is also identical

to Jack and Janet with respect to her earnings, utility, and luck. Chrissy's

probability of good luck is independent from either Jack's or Janet's. After

some discussion, they decide that Chrissy should join the agreement of

Jack and Janet. All three of them will pool their earnings and then split

them equally three ways.

(f)What are the eight possible luck-outcome triplets, and what is the

probability of reaching each of them?

(g)What is the expected utility for each of the investors under this expanded

arrangement?

(h)What amount of certain income would yield the same level of utility

as in part(g)for these risk-averse investors?