This paper constructs a model of saving for retired single people that includes heterogeneity in medical expenses and life expectancies, and bequest motives. We estimate the model using AHEAD data and the method of simulated moments. Out-of-pocket medical expenses rise quickly with age and permanent income. The risk of living long and requiring expensive medical care is a key driver of saving for many higher income elderly. Social insurance programs such as Medicaid rationalize the low asset holdings of the poorest, but also benefit the rich, by insuring them against high medical expenses at the ends of their lives.