This paper considers a life-time consumption-investment problem under the Black-Scholes framework, where the investors consumption rate is subject to a lower bound constraint that linearly depends on the investors wealth. Due to the state-dependent control constraint, the standard stochastic control theory cannot be directly applied to our problem. We overcome this obstacle by examining an equivalent problem that does not impose state-dependent control constraint. It is shown that the value function is a third-order continuously differentiable function by using differential equation approaches. The feedback form optimal consumption and investment strategies are given. According to our findings, if the investor is more concerned with long-term consumption than short-term consumption, then she should, regardless of her financial condition, always consume as few as possible; otherwise, her optimal consumption strategy is state-dependent: consuming optimally when her financial condition is good, and consuming at the lowest possible rate when her financial situation is bad.
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