b(\tilde{t}_{1})} \left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} \frac{F_{1}(t)}{V_{1}(t ; v_{1})^{2}}, \\
\frac{\partial^{2}}{\partial V_{1}(t ; v_{1})^{2}} \left(\hat{\pi}_{1}(t ; v_{1})\right)_{i} = {} & - 2 \frac{1}{1 - b(\tilde{t}_{1})} \left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} \frac{F_{1}(t)}{V_{1}(t ; v_{1})^{3}}.\end{aligned}$$ Let ${\left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} > 0}$ for ${i \in \left\{1, \hdots, N\right\}}$, then

1.  ${\frac{\partial}{\partial V_{1}(t ; v_{1})} \left(\hat{\pi}_{1}(t ; v_{1})\right)_{i} \stackrel{(>)}{\ge} 0\ \Leftrightarrow\ F_{1}(t) \stackrel{(>)}{\ge} 0}$.

2.  ${\frac{\partial^{2}}{\partial V_{1}(t ; v_{1})^{2}} \left(\hat{\pi}_{1}(t ; v_{1})\right)_{i} \stackrel{(<)}{\le} 0\ \Leftrightarrow}$ either ${F_{1}(t) \stackrel{(>)}{\ge} 0}$ and ${V_{1}(t ; v_{1}) > 0}$ or ${F_{1}(t) \stackrel{(<)}{\le} 0}$ and ${V_{1}(t ; v_{1}) < 0}$.

This implies that at time $t$:

1.  $\left(\hat{\pi}_{1}(t ; v_{1})\right)_{i}$ is increasing in $V_{1}(t ; v_{1})$ if and only if ${F_{1}(t) \ge 0}$, and decreasing in $V_{1}(t ; v_{1})$ otherwise.

2.  $\left(\hat{\pi}_{1}(t ; v_{1})\right)_{i}$ is concave in $V_{1}(t ; v_{1})$ if and only if

    1.  either ${F_{1}(t) \ge 0}$ and ${V_{1}(t ; v_{1}) > 0}$

    2.  or ${F_{1}(t) \le 0}$ and ${V_{1}(t ; v_{1}) < 0}$,

    and convex in $V_{1}(t ; v_{1})$ otherwise.

The opposite inequalities and conclusions for $\left(\hat{\pi}_{1}(t ; v_{1})\right)_{i}$ and its derivatives apply if ${\left(\Sigma^{-1} (\mu - r \mathbf{1})\right)_{i} < 0}$.

The optimal controls in Theorem \[thm:Solution:ConsumptionOnly\] determine the value function and the value 