 Solution

The stated portfolio has initial wealth $5$ and terminal payoffs $W_u=4\cdot 6-3\cdot 7=3$ and $W_d=4\cdot 6-3\cdot 4=12$. For CRRA utility, $U^{\prime }(w)=w^{-R}$. The market is complete, and an interior optimum in its positive payoff domain has marginal utility proportional to the unique state-price density. ThereforeThe Lagrange multiplier is $y^{\ast }=9/(10\sqrt{3})$, and both state equations $U^{\prime }(W)=y^{\ast }Z$ are satisfied. Part (b) verifies sufficiency and strict concavity gives uniqueness of the optimal payoff. This illustrates risk aversion recovered from an optimal two-state payoff.