We follow the approach by [@Aase2017], likewise for a HARA utility function. We extend the findings of [@Aase2017] by introducing a time-varying, deterministic consumption floor $\bar{c}(t)$, a time-varying preference function $a(t)$ of consumption over terminal wealth and an income-rate process $y(t)$.

In order to guarantee the consumption rate floor, note $c(t) > \bar{c}(t)$, let us assume the following lower boundary for $v_{1}$ which equals the integral over the discounted consumption floor rate minus income rate over the whole horizon of interest: $$\begin{aligned}
\begin{split} \label{eq:Condition:v1_&_eq:def:F1(t)}
v_{1} > {} & \int_{0}^{T} e^{- r s} \left(\bar{c}(s) - y(s)\right) ds =: F_{1}(0), \\
F_{1}(t) := {} & {\mathbb{E}}\left[\int_{t}^{T} \frac{\tilde{Z}(s)}{\tilde{Z}(t)} \left(\bar{c}(s) - y(s)\right) ds \Big| \mathcal{F}_{t}\right] = \int_{t}^{T} e^{- r (s-t)} \left(\bar{c}(s) - y(s)\right) ds. 
\end{split}\end{aligned}$$ Notice that ${v_{1} < 0}$ is possible since a sufficiently large positive income stream can be high enough to finance consumption. Using the martingale method we solve the problem as summarized by the theorem below.

\[thm:Solution:ConsumptionOnly\] The solution to the optimal stochastic control problem with intertemporal utility function $U_{1}$ in is $$\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})}, \\
c_{1}(t;v_{1}) = {} & g(t,t; 