versa. Further note that $a(t)$ does not appear in $\mathcal{A}_{1}(t,c)$. Therefore we have two input functions $a(t)$ and $b(t)$ where $a(t)$ has no influence on risk aversion, but $b(t)$ determines it; hence a very flexible model.

Since we have $c(t) > \bar{c}(t)$ and $V(T) > F$ by definition of the utility functions in , we restrict $$\begin{aligned}
 \label{eq:Condition:v0:minimalrequirement}
v_{0} > \int_{0}^{T} e^{- r s} \left(\bar{c}(s) - y(s)\right) ds + e^{- r T} F =: F(0)\end{aligned}$$ on the initial endowment in . It is useful to define $$\begin{aligned}
F(t) = {} & {\mathbb{E}}\left[\int_{t}^{T} \frac{\tilde{Z}(s)}{\tilde{Z}(t)} \bar{c}(s) ds + \frac{\tilde{Z}(T)}{\tilde{Z}(t)} F - \int_{t}^{T} \frac{\tilde{Z}(s)}{\tilde{Z}(t)} y(s) ds \Big| \mathcal{F}_{t}\right] \nonumber \\
= {} & \int_{t}^{T} {\mathbb{E}}\left[\frac{\tilde{Z}(s)}{\tilde{Z}(t)} \Big| \mathcal{F}_{t}\right] \left(\bar{c}(s) - y(s)\right) ds + F {\mathbb{E}}\left[\frac{\tilde{Z}(T)}{\tilde{Z}(t)} \Big| \mathcal{F}_{t}\right] = \int_{t}^{T} e^{- r (s-t)} \left(\bar{c}(s) - y(s)\right) ds + e^{- r (T-t)} F. \label{eq:def:F(t)}\end{aligned}$$ $F(t)$ can be interpreted as the time $t$ value of all future minimal liabilities less income. $F(t)$ equals the sum of the time $t$ wealth necessarily required to meet all the future minimum living expenses and expenditures $\bar{c}(s)$, ${s \in [t,T]}$ during the remaining time and the time $t$ value of the minimum desired terminal wealth level $F$; future salary income is 