I don’t understand using and the revealed preferences independently of each other. only makes sense if it is consistent with the revealed preferences that people place on health vs income. If revealed preferences show that people have a constant valuation of income doublings vs life, then that is only consistent with , and I see no justification for using . How would you justify it?
Maybe this is where our two approaches differ:
We have three types of valuations.
Health benefits across different levels of
Income benefits across different levels of
Health vs income benefits at each level of
My approach is to estimate from data on people’s choices regarding (2) (e.g., p. 7 here). Then get the health vs income moral weights from revealed preferences on (3) (e.g., VSLY data). Then combine the two to get (1).
What I think you’re saying is (correct me if I’m wrong) that (3) also gives us data on , because individuals value extra life-years proportional to the consumption in that year. (By assuming that empirical VSLY estimates are described by , this gives us info on by choosing the that makes best fit the VSLY revealed preferences.) Based on that, it then is inconsistent to have a different from (2) than is implied by (3), and we want to reconcile them.
Does that sound right? If so, my view would be that valuing an extra life year according to for some is a functional form assumption on how people value an extra life-year. In some way, I see the data on from (2) as a test of that assumption. Whereas in your view, which I think is also reasonable, the assumption and data on (3) is a test/verification of our estimate of from (2).
My earlier statement did not rely explicitly on the VSLY being . However, what it does rely on is the VSLY-income ratio being increasing in . If we assume the value of health is constant and that , then the VSLY is so the VSLY-income ratio is constant. I’m down to assume the value of health is constant, and I don’t feel strongly about even though I think it’s probably . But my loose reading of the VSLY-income literature is that the ratio is increasing in .
If you don’t think that we know about from (3), for example because you don’t believe the assumption, higher can imply higher valuation of health at lower incomes even if the VSLY-income ratio is increasing in . Here is a hypothetical example. Suppose the elasticity is 1.2 so that a $1,000 earner is indifferent between 1 income doubling and an extra life-year, while a $2,000 earner is indifferent between 1.1 income doubling and an extra life-year. That means that at , the value placed on health for the $2,000 earner is higher than for the $1,000 earner. Increase and normalise the moral weights on income doublings so that an income doubling for the $1,000 earner has the same value as before. Higher means the income doubling at $2,000 is now less valuable. Since the VSLY revealed preferences are unchanged, the moral weight on health at higher incomes is now lower relative to the weight on health at lower incomes, compared to before.
I think that assumption isn’t sufficient to determine η from VSLY data. By not specifying the functional form for VSLYs, η will be underidentified in practice. Assuming only that the denominator of the VSLY term is u′(c) and that u(⋅) is isoelastic, we could, for example, have either of the following:
VSLYc=kcu′(c)=kcη−1
VSLYc=u(c)cu′(c)=cη−1−1η−1 or ln(c) if η=1
Now suppose you observe the VSLY/income data and think it’s roughly ln(c). Would you conclude from this that η=1 and the right functional form is (2)? Or that η=1.2 and the right functional form is (1) with k=1.8? (Plot)
Does that sound right to you? If so, I think that puts us here:
______
One counterargument might be that for any of these functional forms, an increase in η will lead to higher valuations of health at higher incomes. However, I’m not sure that works in practice. I’d imagine it going something like this (assuming the only assumption we’re willing to make ex-ante is that the VSLY denominator is u′(c)):
Estimate η from data on consumption valuations at different income levels
Choose VSLY functional form to fit the revealed preferences data given the η estimate from (1).
Revise our estimate in (1) upwards.
Since we’re still trying to match the VSLY data given the η from 1: adjust the VSLY functional form. (Rather than increasing η in the previous form)