You’re pointing out that the value of life shouldn’t depend on consumption u(c). But even if you assumed there was a fixed utility from saving lives, the VSLY would still be different across populations, and all of the conclusions I talked about would still hold. The reason is because of the monetization of utility. The VSLY is grounded in how an individual would trade off health vs income, and that tradeoff depends on their marginal utility of income u′(c).
Assume there was a fixed utility from someone being alive for a year, ¯u. Then their willingness to pay for an extra year of life is ¯u/u′(c). This varies less with income than if the value of life was u(c), but it still varies with income. Poor people are willing to spend less (in absolute terms) on health than rich people are, because their competing needs are higher than rich people’s competing needs.
If this is the case, then all of our discussion of η still goes through. For an isoelastic utility function, the VSLY is ¯u⋅cη, which increases in income, and increases exponentially in η. The replacement of u(c) with ¯u changes nothing qualitatively; higher η still leads to higher VSLYs even when everyone enjoys a year of life equally. That then causes health interventions for the rich to dominate health interventions for the poor.
The reason why this matters is because it renders the following statement problematic:
Take again the purely utilitarian view that we should value utility increases the same irrespective of the beneficiary. Then, note that there is some fixed number x
of income doublings such that people are indifferent between them and an extra year of life, regardless of income level (assuming the VSL-income elasticity is 1).
Assuming the VSL-income elasticity is 1 is equivalent to assuming η=1, since you have to assume that income doublings are worth the same to people regardless of their income levels. So I don’t really understand how that assumption helps you draw conclusions about what would happen if η>1. If η>1, then even if the value of life was common to everyone, the VSL-income elasticity would be greater than 1. Basically, the marginal utility of income doublings declines, but the marginal value of life does not, so the VSL has to grow faster than income.
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.
Both VSLY utility models don’t seem to align with people’s reported VSL preferences with regards to the mortality risk vs income gain tradeoff.
I think it fits fine? Under the log utility model with no set point, WTP for an extra year of life is cln(c). Normalize this by income levels c to get the VSLY-income ratio as ln(c). Eyeballing the graph from OP’s collection of VSL estimates, they look vaguely consistent with a logarithmic relationship between the VSLY-income ratio and income. Or take the general isoelastic utility model, where WTP for an extra year of life is $\frac{c^{\eta}-c}{\eta-1}, so the VSLY-income ratio is 1η−1⋅cη−1−1η−1. This is a general sub-linear function of c for η>1, and you could probably fit a value of η to match the estimates best. Robinson et al (2019) take this elasticity approach and find a roughly log-linear relationship between VSLY-income ratio and income.
Now, these studies can’t statistically reject a flat VSLY-income ratio across countries, so you would be on fair ground to assume a constant VSLY-income ratio. But it’s definitely not right to say that the isoelastic utility model doesn’t fit the data on VSLs across countries.
Got it—I think I might have had an errant c term where I was thinking about fit. Point taken that you can model a non-constant VSLY-income ratio with isoelastic utility.
What do you find to be the strongest reason to use VSLY’s to value lives saved at different income levels? My intuitive approach would have been to use the value of utility from one year of consumption directly, not divided by people’s marginal utility of consumption (i.e., u(c) instead of u(c)u′(c)). We would then value extra life-years only based on the utility received during that year. In the constant utility case, for example, we’d place the same value on an additional life-year regardless of income level.
Isn’t comparing interventions by the utility they create (rather than how that utility is monetised) what we want to do? Analogously, we currently think individuals value income doublings the same at all income levels. We wouldn’t then conclude that the fact that a $1,000 earner has a $1000 WTP for an income doubling while the $2,000 earner has a WTP of $2,000 means we should value the income doubling for the richer person twice as much. But it seems like that’s what the VSLY approach is doing, if I’m not misunderstanding it?
What do you find to be the strongest reason to use VSLY’s to value lives saved at different income levels? My intuitive approach would have been to use the value of utility from one year of consumption directly, not divided by people’s marginal utility of consumption (i.e., u(c) instead of u(c)u′(c)). We would then value extra life-years only based on the utility received during that year. In the constant utility case, for example, we’d place the same value on an additional life-year regardless of income level.
“Value” is a slippery term here. You’re referring to utility value, but when we calculate cost-effectiveness, we have to place a monetary value on the outcome, because
Utility functions are not normalized to scale – we could multiply utility functions by 1000 and nothing would change, even though the utility numbers would increase by 1000x. So already “utility” is not a meaningful concept to put into cost-effectiveness calculations.
We anyways have to compare health interventions and income interventions, which requires having some willingness-to-pay for the utility purchased by a health intervention. “We are willing to pay $X to extend life by one year” – that X has to come from some framework. Whether you call it a VSLY, or a moral weight, or a willingness-to-pay, it comes from somewhere.
The choice of VSLY as a source of moral weight comes from the premise that when comparing interventions that increase health or increase income among poor people, we should make the tradeoff in the same way that they would, rather than imposing our own preferences of how to make that tradeoff. The reason why VSLYs satisfy that is because they are either estimated from revealed-preference decisions of the relevant populations, or from the stated preferences of the relevant populations. (If they were just dictated from on high, they would not be good moral grounding!) Specifically, in a neoclassical framework where people can pay for health improvements, their willingness to pay is exactly the marginal utility of a health improvement, divided by the marginal utility of income u′(c). That’s why we divide by u′(c).
Of course, that leads to indeterminacy when comparing interventions between two populations with different c and thus different VSLYs. So when organizations have a single consistent moral weight on health vs income, they are implicitly treating the VSLY as common between all populations. That basically implements what you’re saying (have a common valuation for years of life across countries). But it doesn’t resolve our earlier discussion, since if you grounded your moral weight on lives saved in a VSLY, and you adopted a new η, you would get a new moral weight on lives saved, which would be the first-order determinant of how η affects grantmaking.
I appreciate the back and forth discussion here, thank you! I agree with most/all of your comment. But: is it not true that your earlier statement
The first step is to note that higher values of η decrease the value of saving poor people’s lives compared to rich people’s lives.
relies on using VSLY=u(c)u′(c) values directly to compare the value of health across different levels of c (which I don’t think we should do)?
The following approach is consistent with your last comment (using VSLY for health vs income tradeoffs at any given c) but would lead us to place a higher value on health at lower vs higher income levels (instead of the other way around, which would be the conclusion no one likes).
Ground your moral weights for income doublings across different levels of c in the ratio of utility from income doublings at different levels of c. (This is invariant to positive affine transformations of u(⋅) by the way.[1])
At each level of c, get the moral weight of health from people’s revealed preferences for health vs income doubling tradeoffs, applied to the moral weight for income doublings at that level of c, which we have from (1). Suppose these tradeoffs are constant in the sense that people are indifferent between e.g., 1 income doubling and 1 extra life-year at all levels of c).
Increase η. Now, at some high c, income doublings get less moral weight than before. Since people’s revealed health vs income doubling tradeoffs are unchanged, there is less moral weight on health at higher income levels.
I don’t understand using η and the revealed preferences independently of each other.η 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 η=1, and I see no justification for using η>1. How would you justify it?
My earlier statement did not rely explicitly on the VSLY being u(c)/u′(c). However, what it does rely on is the VSLY-income ratio being increasing in c. If we assume the value of health is constant ¯u and that η=1, then the VSLY is ¯u⋅c 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 η even though I think it’s probably >1. But my loose reading of the VSLY-income literature is that the ratio is increasing in c.
I don’t understand using η and the revealed preferences independently of each other.η 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 η=1, and I see no justification for using η>1. 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 c
Income benefits across different levels of c
Health vs income benefits at each level of c
My approach is to estimate η 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 η, because individuals value extra life-years proportional to the consumption in that year. (By assuming that empirical VSLY estimates are described by u(c)u′(c), this gives us info on η by choosing the η that makes u(c)u′(c) best fit the VSLY revealed preferences.) Based on that, it then is inconsistent to have a different η 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 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).
My earlier statement did not rely explicitly on the VSLY being u(c)/u′(c). However, what it does rely on is the VSLY-income ratio being increasing in c. If we assume the value of health is constant ¯u and that η=1, then the VSLY is ¯u⋅c 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 η even though I think it’s probably >1. But my loose reading of the VSLY-income literature is that the ratio is increasing in c.
If you don’t think that we know about η from (3), for example because you don’t believe the assumption, higher η can imply higher valuation of health at lower incomes even if the VSLY-income ratio is increasing in c. 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 η=1, the value placed on health for the $2,000 earner is higher than for the $1,000 earner. Increase η 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 η 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.
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).
______
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.
You’re pointing out that the value of life shouldn’t depend on consumption u(c). But even if you assumed there was a fixed utility from saving lives, the VSLY would still be different across populations, and all of the conclusions I talked about would still hold. The reason is because of the monetization of utility. The VSLY is grounded in how an individual would trade off health vs income, and that tradeoff depends on their marginal utility of income u′(c).
Assume there was a fixed utility from someone being alive for a year, ¯u. Then their willingness to pay for an extra year of life is ¯u/u′(c). This varies less with income than if the value of life was u(c), but it still varies with income. Poor people are willing to spend less (in absolute terms) on health than rich people are, because their competing needs are higher than rich people’s competing needs.
If this is the case, then all of our discussion of η still goes through. For an isoelastic utility function, the VSLY is ¯u⋅cη, which increases in income, and increases exponentially in η. The replacement of u(c) with ¯u changes nothing qualitatively; higher η still leads to higher VSLYs even when everyone enjoys a year of life equally. That then causes health interventions for the rich to dominate health interventions for the poor.
The reason why this matters is because it renders the following statement problematic:
Assuming the VSL-income elasticity is 1 is equivalent to assuming η=1, since you have to assume that income doublings are worth the same to people regardless of their income levels. So I don’t really understand how that assumption helps you draw conclusions about what would happen if η>1. If η>1, then even if the value of life was common to everyone, the VSL-income elasticity would be greater than 1. Basically, the marginal utility of income doublings declines, but the marginal value of life does not, so the VSL has to grow faster than income.
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.
I think it fits fine? Under the log utility model with no set point, WTP for an extra year of life is cln(c). Normalize this by income levels c to get the VSLY-income ratio as ln(c). Eyeballing the graph from OP’s collection of VSL estimates, they look vaguely consistent with a logarithmic relationship between the VSLY-income ratio and income. Or take the general isoelastic utility model, where WTP for an extra year of life is $\frac{c^{\eta}-c}{\eta-1}, so the VSLY-income ratio is 1η−1⋅cη−1−1η−1. This is a general sub-linear function of c for η>1, and you could probably fit a value of η to match the estimates best. Robinson et al (2019) take this elasticity approach and find a roughly log-linear relationship between VSLY-income ratio and income.
Now, these studies can’t statistically reject a flat VSLY-income ratio across countries, so you would be on fair ground to assume a constant VSLY-income ratio. But it’s definitely not right to say that the isoelastic utility model doesn’t fit the data on VSLs across countries.
Got it—I think I might have had an errant c term where I was thinking about fit. Point taken that you can model a non-constant VSLY-income ratio with isoelastic utility.
What do you find to be the strongest reason to use VSLY’s to value lives saved at different income levels? My intuitive approach would have been to use the value of utility from one year of consumption directly, not divided by people’s marginal utility of consumption (i.e., u(c) instead of u(c)u′(c)). We would then value extra life-years only based on the utility received during that year. In the constant utility case, for example, we’d place the same value on an additional life-year regardless of income level.
Isn’t comparing interventions by the utility they create (rather than how that utility is monetised) what we want to do? Analogously, we currently think individuals value income doublings the same at all income levels. We wouldn’t then conclude that the fact that a $1,000 earner has a $1000 WTP for an income doubling while the $2,000 earner has a WTP of $2,000 means we should value the income doubling for the richer person twice as much. But it seems like that’s what the VSLY approach is doing, if I’m not misunderstanding it?
“Value” is a slippery term here. You’re referring to utility value, but when we calculate cost-effectiveness, we have to place a monetary value on the outcome, because
Utility functions are not normalized to scale – we could multiply utility functions by 1000 and nothing would change, even though the utility numbers would increase by 1000x. So already “utility” is not a meaningful concept to put into cost-effectiveness calculations.
We anyways have to compare health interventions and income interventions, which requires having some willingness-to-pay for the utility purchased by a health intervention. “We are willing to pay $X to extend life by one year” – that X has to come from some framework. Whether you call it a VSLY, or a moral weight, or a willingness-to-pay, it comes from somewhere.
The choice of VSLY as a source of moral weight comes from the premise that when comparing interventions that increase health or increase income among poor people, we should make the tradeoff in the same way that they would, rather than imposing our own preferences of how to make that tradeoff. The reason why VSLYs satisfy that is because they are either estimated from revealed-preference decisions of the relevant populations, or from the stated preferences of the relevant populations. (If they were just dictated from on high, they would not be good moral grounding!) Specifically, in a neoclassical framework where people can pay for health improvements, their willingness to pay is exactly the marginal utility of a health improvement, divided by the marginal utility of income u′(c). That’s why we divide by u′(c).
Of course, that leads to indeterminacy when comparing interventions between two populations with different c and thus different VSLYs. So when organizations have a single consistent moral weight on health vs income, they are implicitly treating the VSLY as common between all populations. That basically implements what you’re saying (have a common valuation for years of life across countries). But it doesn’t resolve our earlier discussion, since if you grounded your moral weight on lives saved in a VSLY, and you adopted a new η, you would get a new moral weight on lives saved, which would be the first-order determinant of how η affects grantmaking.
I appreciate the back and forth discussion here, thank you! I agree with most/all of your comment. But: is it not true that your earlier statement
relies on using VSLY=u(c)u′(c) values directly to compare the value of health across different levels of c (which I don’t think we should do)?
The following approach is consistent with your last comment (using VSLY for health vs income tradeoffs at any given c) but would lead us to place a higher value on health at lower vs higher income levels (instead of the other way around, which would be the conclusion no one likes).
Ground your moral weights for income doublings across different levels of c in the ratio of utility from income doublings at different levels of c. (This is invariant to positive affine transformations of u(⋅) by the way.[1])
At each level of c, get the moral weight of health from people’s revealed preferences for health vs income doubling tradeoffs, applied to the moral weight for income doublings at that level of c, which we have from (1). Suppose these tradeoffs are constant in the sense that people are indifferent between e.g., 1 income doubling and 1 extra life-year at all levels of c).
Increase η. Now, at some high c, income doublings get less moral weight than before. Since people’s revealed health vs income doubling tradeoffs are unchanged, there is less moral weight on health at higher income levels.
Take u(2x2)−u(x2)u(2x1)−u(x1) and v=a+bu. Sub in v for u and the term stays the same.
Likewise, great discussion!
I don’t understand using η and the revealed preferences independently of each other.η 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 η=1, and I see no justification for using η>1. How would you justify it?
My earlier statement did not rely explicitly on the VSLY being u(c)/u′(c). However, what it does rely on is the VSLY-income ratio being increasing in c. If we assume the value of health is constant ¯u and that η=1, then the VSLY is ¯u⋅c 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 η even though I think it’s probably >1. But my loose reading of the VSLY-income literature is that the ratio is increasing in c.
Maybe this is where our two approaches differ:
We have three types of valuations.
Health benefits across different levels of c
Income benefits across different levels of c
Health vs income benefits at each level of c
My approach is to estimate η 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 η, because individuals value extra life-years proportional to the consumption in that year. (By assuming that empirical VSLY estimates are described by u(c)u′(c), this gives us info on η by choosing the η that makes u(c)u′(c) best fit the VSLY revealed preferences.) Based on that, it then is inconsistent to have a different η 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 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).
If you don’t think that we know about η from (3), for example because you don’t believe the assumption, higher η can imply higher valuation of health at lower incomes even if the VSLY-income ratio is increasing in c. 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 η=1, the value placed on health for the $2,000 earner is higher than for the $1,000 earner. Increase η 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 η 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.
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:
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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.