consumption is eating up the whole income, then $$\begin{aligned}
\hat{\pi}_{1}(t ; v_{1}) = \frac{1}{1 - b} \Sigma^{-1} (\mu - r \mathbf{1}),\end{aligned}$$ which is a constant mix strategy and represents the standard, well-known result for CRRA utility with constant risk aversion parameter.

Some comments on the initial capital $v_{1}$ and the sign of the risky investments come next. As already pointed out, a start with a negative initial capital ${V_{1}(0 ; v_{1}) = v_{1} < 0}$ to Problem is possible and might be reasonable in a sense that accumulated income over the life-cycle is expected to exceed total consumption. Hence, there is no need to require positive capital to this problem. For this reason, ${V_{1}(t ; v_{1}) < 0}$ can happen and might be reasonable, too.

Theorem \[thm:Solution:ConsumptionOnly\] tells that the optimal relative investment strategy is given by $$\begin{aligned}
\hat{\pi}_{1}(t ; v_{1}) = {} & \frac{1}{1 - b(\tilde{t}_{1})} \Sigma^{-1} (\mu - r \mathbf{1}) \frac{V_{1}(t ; v_{1}) - F_{1}(t)}{V_{1}(t ; v_{1})},\end{aligned}$$ where ${V_{1}(t ; v_{1}) > F_{1}(t)}$ a.s.. Let ${\left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} > 0}$ for ${i \in \left\{1, \hdots, N\right\}}$, which for instance is the case when there is only one risky asset ($N = 1$) because then ${\Sigma^{-1} \left(\mu 