In a previous “helpful” hint pertaining to the second problem in Problem Set 4, I noted (among other things) that expected utility for Investor A is *E*(*U*(*W*)) = .6(1,020 + 100x)^{.5} + .4(1,020 – 140x)^{.5}, where x corresponds to the proportion of the portfolio that is to be allocated to the risky asset. “Optimal” exposure to risk is determined by maximizing *E*(*U*(*W*)); this is accomplished by differentiating *E*(*U*(*W*)) with respect to x, setting that result equal to 0, and solving for x. The mathematical logic applied here is the same as the approach shown on pp. 14-15 in the Mathematics Tutorial for determining the profit-maximizing production decision of a firm. Here, since the square root utility function is itself a function of x; i.e., *U*(*W*(x)), this means that we must apply the chain rule in order to differentiate *E*(*U*(*W*)) with respect to x:

Since there are two states, this means that the are two state-contingent values for *W*(*x*); specifically, 60% of the time, *W*(*x*) = 1,020 + 100x , and 40% of the time, *W*(*x*) = 1,020 -140x. Once the chain rule has been applied to differentiating both terms on the right-hand side of the *E*(*U*(*W*)) equation for Investor *A*, set that result (also known as the “first-order condition”) equal to 0 and solve for Investor A’s optimal exposure to the risky asset. Once that proportion has been determined, the allocation to the safe asset is equal to 1-x.

Rinse and repeat to determine Investor B’s optimal exposure to risk. Since Investor B’s utility function is *E*(*U*(*W*)) = .6ln (1,020 + 100x) + .4ln(1,020 – 140x), it follows that the first-order condition is Set Investor B’s first-order condition equal to 0, solve for x and 1-x, and you’re good to go!