Looking at when health expenses are made, in particular out-of-pocket ones, a large share of them are made towards the end of one's life, in particular for nursing home stays. Thus when studying savings in an economy, and in particular savings for retirement, it is critical to understand well savings for the nursing home. Of great importance is the institutional context: who pays for the nursing home. In the United States, there is no significant private insurance for nursing home costs, however, Medicaid picks up the tab if some means tests are met. This leads to perverse effects wherein retirees divest their savings anticipating the need for a nursing home.
Karen Kopecky and Tatyana Koreshkova study this in more detail with a life cycle model with uncertainty. They notice a splitting of the population: the rich, whose savings behavior is driven by the large and persistent nursing home costs, and the poor, who only save towards the smaller medical costs (and avoid getting too rich to avoid losing Medicaid benefits).
Introducing generalized public health care obviously reduces the incentives to save, their is less uncertainty. The reduction in saving is drastic for the richer households if one keeps the interest constant. But with lower savings, the latter must increase, which induce more savings. In general equilibrium, the decline in aggregate savings is modest, but welfare is enhanced, due to the dramatic reduction in uncertainty for households who do not face the risk of bankruptcy due to health shocks. Doing the opposite (removing all Medicaid benefits) would dramatically increase aggregate savings (+134%), almost all on account of the poor households. But again, higher savings do not necessarily means households are better off.