Yup, that’s an accurate summary of my beliefs (with the caveat that u(c) is non-critical and can be replaced with a constant or whatever else you want; only u′(c) is essential). Put another way, η is a single preference parameter that determines the marginal utility of income, and that affects how we value both income and health. I think any other assumption leads to internal inconsistency, or doesn’t represent utility maximization.
Does that sound right? If so, my view would be that valuing an extra life year according to u(c)/u′(c) for some η is a functional form assumption on how people value an extra life-year. In some way, I see the data on η 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 η from (2).
What is an individual willing to pay for anything? Suppose buying a health good (e.g. air purifier) gives you utility k, and it costs p. Then for every dollar you spend, you are getting k/p utility. Is that optimal? It is optimal only if the marginal utility of spending a dollar on any other good is ≤k/p. If you could get >k/p utility from spending a dollar elsewhere, then you would optimally refuse to buy the air purifier and spend your money elsewhere. That’s why u′(c) is always in the denominator; it represents the opportunity cost of money. If you didn’t spend your money on a health good, you would spend it somewhere else. And the opportunity cost of money is obviously higher for poor people (trading off daily food for an air purifier is a hell of a lot less appealing than trading off a Chanel bag for an air purifier). So that’s why I don’t think there can be any consistent model of utility maximization where the VSLY[1] doesn’t depend on u′(c). Its dependence on u(c) is irrelevant and can be replaced with some constant k if you want, but I am reasonably sure that u′(c) can’t be banished from the VSLY without rejecting individual utility maximization.
So I think the crux between us is whether you see your position as consistent with VSLYs being derived from individual utility maximization. If it is, then please help me understand, because that would be a major update for me. But if it’s not, then I think we are at this point:
Now, the key assumption is that we monetize VSLs using individual willingness-to-pay. Maybe you think social willingness-to-pay should be determined by the marginal utility of money to the social planner, which is common across people, rather than by the WTP of individuals who vary in their income levels. This is a defensible normative position. I would just note that the marginal utility of money from a donor’s perspective is the value you could otherwise get by spending that money. For us, that benchmark use of money is GiveDirectly cash transfers. If you think that way, you will end up with a marginal utility of money that is close to a poor person’s marginal utility of money, so the original framework is still representative of how valuable health interventions are among poor people.
If you substitute “buying an air purifier” with “buying a year of life”, then my argument goes from estimating “willingness to pay for an air purifier” to estimating “willingness to pay for an extra year of life”. This is exactly what the VSLY represents, when it is estimated from individual revealed preferences.
Yup, that’s an accurate summary of my beliefs (with the caveat that u(c) is non-critical and can be replaced with a constant or whatever else you want; only u′(c) is essential).
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:
If so, my view would be that valuing an extra life year according to u(c)u′(c) for some η is a functional form assumption on how people value an extra life-year. In some way, I see the data on η 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 η from (2).
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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)
I’m agnostic on the right functional form for the VSLY, just as I’m agnostic on the right η. My point was just that you cannot have it be independent of u′(c).
You need to impose some structure to get an exact identification of η, but that should not be interpreted as meaning that we can be fully agnostic about how η affects valuations, the way you describe. So I don’t think that puts us at the point you stated. Specifically, I think the framework you describe where the VSLY relative to income doublings is constant while you shift η is still inconsistent with utility maximization, and still not a valid way to interpret how η affects the value of health vs income.
Yup, that’s an accurate summary of my beliefs (with the caveat that u(c) is non-critical and can be replaced with a constant or whatever else you want; only u′(c) is essential). Put another way, η is a single preference parameter that determines the marginal utility of income, and that affects how we value both income and health. I think any other assumption leads to internal inconsistency, or doesn’t represent utility maximization.
What is an individual willing to pay for anything? Suppose buying a health good (e.g. air purifier) gives you utility k, and it costs p. Then for every dollar you spend, you are getting k/p utility. Is that optimal? It is optimal only if the marginal utility of spending a dollar on any other good is ≤k/p. If you could get >k/p utility from spending a dollar elsewhere, then you would optimally refuse to buy the air purifier and spend your money elsewhere. That’s why u′(c) is always in the denominator; it represents the opportunity cost of money. If you didn’t spend your money on a health good, you would spend it somewhere else. And the opportunity cost of money is obviously higher for poor people (trading off daily food for an air purifier is a hell of a lot less appealing than trading off a Chanel bag for an air purifier). So that’s why I don’t think there can be any consistent model of utility maximization where the VSLY[1] doesn’t depend on u′(c). Its dependence on u(c) is irrelevant and can be replaced with some constant k if you want, but I am reasonably sure that u′(c) can’t be banished from the VSLY without rejecting individual utility maximization.
So I think the crux between us is whether you see your position as consistent with VSLYs being derived from individual utility maximization. If it is, then please help me understand, because that would be a major update for me. But if it’s not, then I think we are at this point:
If you substitute “buying an air purifier” with “buying a year of life”, then my argument goes from estimating “willingness to pay for an air purifier” to estimating “willingness to pay for an extra year of life”. This is exactly what the VSLY represents, when it is estimated from individual revealed preferences.
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)
I’m agnostic on the right functional form for the VSLY, just as I’m agnostic on the right η. My point was just that you cannot have it be independent of u′(c).
You need to impose some structure to get an exact identification of η, but that should not be interpreted as meaning that we can be fully agnostic about how η affects valuations, the way you describe. So I don’t think that puts us at the point you stated. Specifically, I think the framework you describe where the VSLY relative to income doublings is constant while you shift η is still inconsistent with utility maximization, and still not a valid way to interpret how η affects the value of health vs income.