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
(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?