In a previous “helpful” hint pertaining to the first problem in Problem Set 4, I noted (among other things) that expected utility for Investor A is . Since thee square root utility function is itself a function of ; i.e., this means that we must apply the chain rule in order to compute the first order condition and solve for :
. Applying the chain rule, the first order condition for Investor A is:
In order to determine Investor A’s optimal risk exposure, all that remains to be done is to solve this equation for .
Rinse and repeat to determine Investor B’s optimal exposure to risk. Since Investor B’s utility function is , B’s expected utility is . Applying the chain rule to , it follows that . Given these clues, you should be able to determine optimal for Investor B. However, before you begin, keep in mind that since Investor A’s Arrow-Pratt coefficient is and Investor B’s Arrow-Pratt coefficient is (see pages 8, 10, and 12 of http://fin4335.garven.com/fall2019/lecture6.pdf to see how these measures were determined), we know that Investor B is more risk averse than Investor A. Therefore, Investor B will select a lower level of exposure to risk then Investor B.