In: Economics
An umbrella salesman expects profits of $1000 if the summer is rainy, and -$500 if it is sunny. The summer will be rainy with probability p. He can also (if he wishes) invest any amount $K in a sunglasses store, which pays a return rate of rR < 0 if the summer is rainy, rS > 0 if the summer is sunny.
(a) Write out his expected utility if initial wealth is w0, he invests $K in the sunglasses store, and utility from wealth w is u(w), via:
i. Calculate his final wealth, wrainy , in a rainy summer. (Hint: his umbrella business earns $1000 in this case, and the $K in the sunglasses store earns a profit of $K times the rainy-summer return rate).
ii. Calculate his final wealth, wsunny, in a sunny summer.
iii. Use your answers to (i) and (ii) to find expected utility.
(b) If rR = -(1-p) and rS = p, show that a risk-averse umbrella will choose $K so that final wealth does not depend on weather, i.e. wrainy = wsunny .
(c) What investment $K achieves the outcome in (b) (wealth independent of weather), and what is the variance of his total wealth?
*EDIT* I asked this question already, but I miswrote "-$500" as just "$500." Thank you.