 Solution

Assume $S_0\ne 0$ and keep $d=S_0^{T}M{S}_0>0$, $\eta =(S_0^{T}M\mu -w_0)/d$. The constraint $x\ge 0$ is $S_0^T\theta \le w_0$. If the unrestricted optimum in (i) lies in this half-space, it remains optimal. Its cash holding isThus it has strictly positive cash precisely when $\eta <R$. If $\eta >R$, the free optimum violates the constraint, so strict concavity and the Karush-Kuhn-Tucker conditions put the optimum on the boundary, where it is the cash-free solution in (ii). At equality the two solutions coincide. ThereforeIn particular,The multiplier for the inequality is $\max (0,\eta -R)$, explicitly checking the complementary-slackness condition.