\frac{1-\hat{b}}{\hat{b}}    \left(\frac{1}{1-\hat{b}}  (v-F)\right)^{\hat{b}},
\end{split}\end{aligned}$$ for $\beta \ge 0$, $b : [0,T] \to (- \infty, 1)\backslash \{0\}$ continuous, $\hat{b} < 1$, $\hat{b} \neq 0$, $a(t) > 0$, $\hat{a} > 0$, $c(t) > \bar{c}(t)$, $\bar{c}(t) \ge 0$ deterministic, and $v > F$ with $F \ge 0$. $U_{2}$ is a continuously differentiable and strictly concave terminal utility function, $U_{1}$ denotes a continuous (intertemporal consumption) utility function which is continuously differentiable and strictly concave in the second argument. This utility model accounts for several desired aspects: minimum liquid asset wealth level $F \ge 0$ at the age of retirement $T$, minimum consumption rate $\bar{c}(t) \ge 0$ and time-varying preference of consumption over terminal wealth in terms of $a(t)$. Moreover, the coefficient of risk aversion $b(t)$ in the consumption utility is now a continuous function in time.

Notice that the associated *Arrow-Pratt measure* ${\mathcal{A}(v) := - \frac{U''(v)}{U'(v)} = - \frac{\partial}{\partial v} \ln U'(v)}$ of absolute risk aversion, developed by [@Pratt1964] and [@Arrow1970], admits the following hyperbolic representation $$\begin{aligned}
\mathcal{A}_{1}(t,c) = \frac{1 - b(t)}{c - \bar{c}(t)},\ \mathcal{A}_{2}(v) = \frac{1 - \hat{b}}{v - F}.\end{aligned}$$ For this reason, we use the notation of an increasing $b(t)$ as a synonym for a decreasing coefficient of risk aversion and vice 