This paper provides a detailed study of a simple life-cycle consumption model with quasi-hyperbolic discounting and an imperfect financial market. It gives a complete characterization of savings behaviors. The joint assumptions of quasi-hyperbolic discount factors and no-borrowing constraints may lead to non-convexities in selves' objective functions that may imply discontinuous equilibrium strategies. Savings function may undergo jumps and non-momotonicities when the income or the interest rate reach a threshold value. These "anomalies" may exist even for reasonable parameters values.