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 behavior. 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. The savings function may undergo jumps and non-monotonicities when the income or the interest rate reaches a threshold value. These ''anomalies'' may exist even for reasonable parameter values.