= Solution
The stated <portfolio> has initial wealth $5$ and terminal payoffs $W_u=4\cdot6-3\cdot7=3$ and $W_d=4\cdot6-3\cdot4=12$. For <CRRA utility>, $U'(w)=w^{-R}$. The market is complete, and an interior optimum in its positive payoff domain has marginal utility proportional to the unique <state-price density>. Therefore
$$
\frac{3^{-R}}{12^{-R}}=\frac{10/9}{5/9}=2,\qquad4^R=2,\qquad\boxed{R=\frac12.}
$$
The <Lagrange multiplier> is $y^*=9/(10\sqrt3)$, and both state equations $U'(W)=y^*Z$ 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>.
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