= Solution
Assume $S_0\ne0$ and keep $d=S_0^TMS_0>0$, $\eta=(S_0^TM\mu-w_0)/d$. The constraint $x\geq0$ is $S_0^T\theta\leq w_0$. If the unrestricted optimum in (i) lies in this half-space, it remains optimal. Its cash holding is
$$
x_{\rm free}=w_0-S_0^TM\mu+Rd=d(R-\eta).
$$
Thus 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. Therefore
$$
\boxed{\theta^*=M[\mu-\max(R,\eta)S_0],\qquad x^*=\max(0,w_0-S_0^TM\mu+Rd).}
$$
In particular,
$$
\boxed{x^*=0\iff\frac{S_0\cdot(\gamma V)^{-1}\mu-w_0}{S_0\cdot(\gamma V)^{-1}S_0}\geq1+r.}
$$
The multiplier for the inequality is $\max(0,\eta-R)$, explicitly checking the complementary-slackness condition.
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