This contribution is an attempt to analyze an issue that seems to be frequently skirted over when discussing reforms of today's health care systems. Practitioners of health insurance point out the fact that a major share of medical care services goes to the uppermost age brackets, who are believed to be unresponsive to the financial incentives (such as cost sharing and bonuses for no claims) that are being considered as innovations in the financing of health care. At the same time, rapid growth in the numbers of the old and very old is predicted for virtually all OECD countries. Acting in combination, these two factors appear to undercut the rationale of conventional cost containment strategies in health care. A closer look at the matter even reveals that a major part of lifetime health resources are spent during the last few months of human life. Economists are therefore challenged to explain a puzzle: Why is it that such a great deal of resources is dedicated to investments that will in all likelihood be characterized by a very short payoff period? In an attempt at answering this question, the decision situations facing the dying patient and his family, treating physicians, and politicians acting as agents of the population at large will be examined in turn. Of course, no easy way out can be expected to result from this exercise. Nevertheless, economic analysis can be counted upon to provide at least cues concerning the incentives that have to be built into future insurance contracts, physician payment schemes, and modes of hospital Finance, for dealing with the challenge to the health care system posed by a rapidly ageing population.