A few days ago Greg Mankiw posted links on his blog to Robert Fogel's recent piece in The American on health care, and Daron Acemoglu's two cents about it. I think Mankiw hit the nail on the head with the confounding role of technological change in the income elasticity. It is definitely an issue of whether we're talking about the long-term, in which technology is changing, or cross-section, in which it isn't, and Fogel and others clearly mean the former.
But the former is arguably what's more relevant if we're concerned about the long-term sustainability of our system of health insurance and health care, using the rising share of health spending in GDP over time as our main indicator. In the first brush anyway, we're less concerned about how much more money is spent on health care by rich as opposed to poor consumers, although that is also an interesting paper.
Acemoglu and co's paper is great in that it totally takes the bait laid by Hall and Jones (QJE, 2007) about microstudies that could shed light on the income elasticity of life extension --- and upends the result. But the inconsistent results remind me of the very similar disagreement in the empirical literature about the income elasticity of the value of a statistical life, which also seems to break along longitudinal vs. cross-sectional lines. (Acemoglu et al. have 2 decades of data, but even that may be relatively short.) Viscusi and Aldy (J Risk Uncertainty, 2003) effectively examine a global cross section of countries after 1970; Costa and Kahn (J Risk Uncertainty, 2004) look at four decades of data in the U.S. The former believe the income elasticity is less than 1; the latter believe it's greater than 1.
I work in a related area, and the latest I heard from Chad Jones about this issue is that their model requires the elasticity of intertemporal substitution, which governs how "luxury" a good life extension is relative to consumption, needs to be less than one for there to be demand-driven growth in the health share of GDP, to paraphrase Fogel. In other words, for the longitudinal evidence and theory to be right.