Tyler Cowan alerted me to this NBER Working Paper by Sherry Glied. (Unfortunately, the full report is gated.) Based on an analysis of data from 20 developed countries, she concludes:

1. There is no general relationship between the way in which countries pay for health care and their ability to control costs. Public v. private financing, general revenue v. payroll taxes, third-party v. out-of-pocket spending - nothing seems to matter very much.
2. Government provision of health care is only modestly progressive. In Canada, people in the bottom two income quintiles - with 40% of the population - get about 50% of the health care benefits. Moreover, relative to health care needs, Canada's health care spending may not be progressive at all. For OECD countries generally, among people with similar health conditions, "higher income people use the system more intensively and use more costly services than do lower income people."
3. Marginal increases in health care spending may actually be regressive. This is especially true if extra spending buys specialist services and elective procedures. "In Canada, high income people make disproportionate use of elective surgical procedures, such as hip and knee replacements."
4. Government provision of health care has little impact on the distribution of well-being in society. When economists assign a monetary value to health care and add it to money income, national health insurance has very little impact on overall economic inequality.
5. Increases in health care spending crowd out other government spending. Redistribution through government-funded health care partly replaces other redistributive government programs. What low-income people gain in health services may be offset by reductions, say, in housing or education benefits.

Here is the bottom line: "A mixed financing system [i.e., one that combines public and private insurance] may be the optimal way to balance efficiency and equity in health care."