as a durable good. Constraints in the optimization problem are considered in [@Cuoco1997], [@Elie2008] and [@Grandits2015]. Moreover, [@Akian1996], [@Altarovici2017] and [@Dai2009] analyze the portfolio problem under transaction costs. The application of HARA utility functions in a life-cycle context can be found in [@Huang2008], [@Ye2008], [@ChangRong2014], [@ChangChang2017] and [@Wang2017]. Moreover, [@BackLiuTeguia2019] study a life-cycle consumption problem for HARA utility with time-independent, increasing risk aversion and examine the relation between age and portfolio risk by using Monte Carlo analysis. [@TangPurcalZhang2018] study an optimal consumption-investment problem under CRRA utility function with age-independent risk aversion, but examine the impact of hyperbolic discounting, where the rate of time preference is a function of time. We generalize this approach by considering general $a(t)$ or ${e^{- \beta t} a(t)}$, respectively, and by introducing age-varying risk aversion.

In this paper we apply HARA utility functions on both the consumption and terminal wealth and consider time-varying preferences: an age-depending preference between consumption and terminal wealth and an age-depending coefficient of risk aversion in the intertemporal consumption utility. For simplicity, income is treated as a deterministic process. Furthermore, we do not model mortality and consider a fixed time horizon $T$ that corresponds to a retirement age, thus we assume the 