Yup, that's an accurate summary of my beliefs (with the caveat that $u\left(c\right)$ is non-critical and can be replaced with a constant or whatever else you want; only $u^{\prime }\left(c\right)$ is essential).

I think that assumption isn't sufficient to determine $\eta$ from VSLY data. By not specifying the functional form for VSLYs, $\eta$ will be underidentified in practice. Assuming only that the denominator of the VSLY term is $u^{\prime }\left(c\right)$ and that $u(\cdot )$ is isoelastic, we could, for example, have either of the following:

1. $\frac{VSLY}{c}=\frac{k}{c{u}^{\prime }\left(c\right)}=k{c}^{\eta -1}$ 
2. $\frac{VSLY}{c}=\frac{u\left(c\right)}{c{u}^{\prime }\left(c\right)}=\frac{c^{\eta -1}-1}{\eta -1}$ or $\ln \left(c\right)$ if $\eta =1$ 

 Now suppose you observe the VSLY/income data and think it's roughly $\ln \left(c\right)$. Would you conclude from this that $\eta =1$ and the right functional form is (2)? Or that $\eta =1.2$ 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 $\frac{u\left(c\right)}{u^{\prime }\left(c\right)}$ for some $\eta$ is a functional form assumption on how people value an extra life-year. In some way, I see the data on $\eta$ 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 $\eta$ from (2).

______

One counterargument might be that for any of these functional forms, an increase in $\eta$ 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 $u^{\prime }\left(c\right)$):

1. Estimate $\eta$ from data on consumption valuations at different income levels
2. Choose VSLY functional form to fit the revealed preferences data given the $\eta$ estimate from (1).
3. Revise our estimate in (1) upwards.
4. Since we're still trying to match the VSLY data given the $\eta$ from 1: adjust the VSLY functional form. (Rather than increasing $\eta$ in the previous form)

I don't understand using $\eta$ and the revealed preferences independently of each other. $\eta$ 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 $\eta =1$, and I see no justification for using $\eta >1$. How would you justify it?

Maybe this is where our two approaches differ:

We have three types of valuations.

1. Health benefits across different levels of $c$
2. Income benefits across different levels of $c$
3. Health vs income benefits at each level of $c$

My approach is to estimate $\eta$ 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 $\eta$, because individuals value extra life-years proportional to the consumption in that year. (By assuming that empirical VSLY estimates are described by $\frac{u\left(c\right)}{u^{\prime }\left(c\right)}$, this gives us info on $\eta$ by choosing the $\eta$ that makes $\frac{u\left(c\right)}{u^{\prime }\left(c\right)}$ best fit the VSLY revealed preferences.) Based on that, it then is inconsistent to have a different $\eta$ 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 $\frac{u\left(c\right)}{u^{\prime }\left(c\right)}$ for some $\eta$ is a functional form assumption on how people value an extra life-year. In some way, I see the data on $\eta$ 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 $\eta$ from (2).

My earlier statement did not rely explicitly on the VSLY being $u\left(c\right)/u^{\prime }\left(c\right)$. However, what it does rely on is the VSLY-income ratio being increasing in $c$. If we assume the value of health is constant $\overline{u}$ and that $\eta =1$, then the VSLY is $\overline{u}\cdot c$ 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 $\eta$ 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 $\eta$ from (3), for example because you don't believe the assumption, higher $\eta$ 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 $\eta =1$, the value placed on health for the $2,000 earner is higher than for the $1,000 earner. Increase $\eta$ 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 $\eta$ 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 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 $\eta$ decrease the value of saving poor people's lives compared to rich people's lives.

relies on using $VSLY=\frac{u\left(c\right)}{u^{\prime }\left(c\right)}$ 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).

1. 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 $u(\cdot )$ by the way.^[1])
2. 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 $c$).
3. Increase $\eta$. 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.

1. ^{^}

   Take $\frac{u\left(2x_2\right)-u\left(x_2\right)}{u\left(2x_1\right)-u\left(x_1\right)}$ and $v=a+bu$. Sub in $v$ for $u$ and the term stays the same.

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., $u\left(c\right)$ instead of $\frac{u\left(c\right)}{u^{\prime }\left(c\right)}$). 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?

Hi Joel,

Thank you for outlining what you're doing at CEARCH—I appreciate it. I've put the Layard, Mayraz, and Nickell review on our list of sources to look at as we investigate the right choice of $\eta$ more. As for where $\eta =1.87$ comes from, I saw that Mo already answered that question (thank you!). Let me know if something is still unclear.

-Nico

Thank you, Karthik! I’ll respond briefly to one point about the inconsistency sometimes leading us to value doubling the incomes of richer people more relative to poorer people. The rest of my comment is on the health vs income benefits discussion, which I’ve found very interesting.

On sometimes preferring to double rich people's incomes

I don't think it makes sense to frame this as valuing rich people over poor people. What's happening in this example is favoring benefits in year 1 over benefits in year 2, regardless of a person's income. This is definitely a nitpick, but I think many people's intuitions about time discounting precede intuitions about rich vs poor people's income doublings, so it's more clarifying to frame it that way.

I see where you’re coming from since we use $\eta >1$ only in our time discounting. Here is the opposite view for why the rich vs poor framing might be more natural:

Recall that the only reason we favour benefits in year 1 over year 2 is that recipients are richer in year 2 (assuming there is no uncertainty about the benefits in year 2). Put another way, if we thought recipients had the same income in year 2 as in year 1, we would not prefer the year 1 benefits. We might in our CEAs think about this as time discounting. But really it is discounting richer people’s income doublings, which is then reflected in our time discount rate only because we assume consumption grows at 3% each year absent any intervention. Because we discount richer people’s income benefits less across interventions than we do within an intervention over time, we have situations like these where we value richer people’s income doublings more.
 

On health vs income

It’s important to draw out the implications of the choice of $\eta$ on the health vs income benefits comparison. With the caveat that I’ve not spent much time looking into previous VSLY discussions, I’m not sure the impact of higher $\eta$ is as clear-cut. Specifically, I’m thinking

1. The conclusions in your comment rely on the specific VSLY utility model $\frac{u\left(c\right)}{u^{\prime }\left(c\right)}$
2. Empirical evidence suggests that model might not reflect individuals’ preferences
3. If we base the analysis on those preferences instead, higher $\eta$ can imply a higher value of a mortality reduction intervention at lower income levels (i.e., the opposite conclusion)

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.

Model 1 (isoelastic utility): Take $\eta =1$ and assume that the elasticity of VSL with regards to income is 1 (i.e., people are indifferent between the same multiple of their income and an extra life-year, regardless of absolute income level). Since an income doubling has the same moral weight regardless of income level, so has an extra life-year. However, model 1 values an extra life-year at $c\ast ln\left(c\right)$, which is increasing in $c$. 

Even if we think the elasticity is 1.3 instead of 1 (the upper bound of what Open Philanthropy considers reasonable), the model doesn’t fit well. Take this hypothetical example: if a $1000 earner is indifferent between $1000 and an extra year of life, a person making $2000 would be indifferent at $2300, and so VSLY should increase by a factor of 1.3, too. However, $c\ast ln\left(c\right)$ equals 6,907 at $c=1000$ and 15,201 at $c=2000$, which is not a 30% but a 220% increase. As you pointed out, $\eta >1$ only leads to preferring health interventions for rich vs poor people even more.

Model 2 (isoelastic utility with set point $k$): The same is true for model 2. Take, for example, $\eta =1$. Then, $VSLY=$ $c\ast (k+ln(c\left)\right)$ (or, for $\eta >1$, $k{c}^{\eta }+\frac{c^{\eta }-c}{\eta -1}$). Both are increasing in $c$, while VSL revealed/stated preferences imply it should be roughly constant or even decreasing (see next point). (This is the same reasoning as for model 1, keeping in mind that $k$ drops out in the value of income doublings, which is unchanged: $u\left(2c\right)-u\left(c\right)=(k+\ln (2c\left)\right)-(k+\ln (c\left)\right))=\ln (2)$)

Since the models don’t reflect individuals’ preferences, I'd be hesitant to use them to make normative claims about how to morally value mortality reduction interventions at different income levels. 
 

When we don’t impose the $VSLY=\frac{u\left(c\right)}{u^{\prime }\left(c\right)}$ model but work up from individuals’ preferences, higher $\eta$ might lead to valuing mortality reduction interventions at lower income levels more (i.e., the opposite conclusion):

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). That means that people gain as much utility from an extra year of life as from $x$ income doublings.

Let $\eta =1$. Income doublings have the same moral value regardless of income level, so the same is true for extra life-years.

Now consider instead $\eta >1$. The income doubling generates less utility at higher income levels than at lower income levels. However, at higher income levels, people are still indifferent between the same $x$ income doublings and an extra life-year. That would imply that the utility (and therefore the moral value) from an extra year of life is lower at higher income levels.

Since in that framework, we value mortality-reduction and income interventions more at lower income levels, my expectation is that higher $\eta$ would lead us to prioritising lower-income beneficiaries more than we currently do.

Hi Karthik, 

Thank you for engaging in detail with the post—these are all great points, much appreciated! I’m replying to each one below.
 

1) “Inconsistency doesn't lead to favouring income doublings for the rich; it just leads to inconsistency”

I agree with your argument that we can also conclude that doubling the incomes of poorer people is more valuable than doubling the incomes of richer people. I meant to make the point in the post that this inconsistency can lead us, strangely, to conclude that doubling the incomes of richer people is more valuable than doubling the incomes of poorer people—not that it always does. To make that point clearer and not give the impression that we systematically favour income doublings for the rich, I’ve edited the post to say things like “can lead us to prefer …” instead of “leads us to prefer …”, and referenced your argument in the appendix.

Substantively, I would draw a somewhat stronger conclusion than you: not only that inconsistency “leads to inconsistency” but that it "sometimes leads us to favour income doublings for the rich”.

That is because, in practice, there are situations where the inconsistency will lead us to that conclusion. Take, for example, deworming in Madagascar (~$500 GDP/capita) vs cash transfers in Kenya (~$2k GDP/capita). For simplicity, assume deworming generates income benefits in year 1 and 2 while cash transfers only generate them in year 1. In our CEAs, we assign the same value for income doublings in year 1 of an intervention. Because we set the comparison point there, we value doubling the income of the $500 and $2k earners the same, but value the income doubling of the $502 earner in Madagascar in year 2 less than for the $2000 earner in Kenya.
 

2) “This inconsistency is not analogous to a rate of pure time preference”. Thank you for pointing out that this wasn't clear. I meant it to be an implication of assuming both $\eta =1$ and $\eta >1$ and applying $\eta =1$ in the example: “even if the income % increases are as certain to occur in 48 years as they are now, we value an income doubling 3x more today.” If those two income doublings are worth the same, the discount rate reflects only pure time preference. To clarify this, I’ve changed that sentence to say “Put another way, if we consistently apply the income doublings framework (log-utility), the CEA implies a 2.6% annual discount rate of pure time preference.” (changes italicised)
 

3) “Choices of eta are probably back-filled from moral weights on health vs income”. That’s an important consideration. We want to make sure any change we make to $\eta$ is also consistent with our moral intuitions about health vs income benefits. One key question seems to be how health benefits vs income doublings trade off at different absolute levels of income? E.g., does the rate at which they trade off stay the same? Do we think health benefits are equally valuable at different income levels but income doublings are worth less at higher income levels? I'd be curious about what your intuitions are here. 

On this point in particular:

Higher values of η, combined with low income levels for recipients, would make income-generating interventions much more attractive than health-generating interventions, and shift your portfolio substantially towards income-generating interventions.

I'm not sure this is necessarily true? It seems like higher values of $\eta$ would primarily increase the slope of the marginal rate of substitution between health benefits and income doublings, as a function of absolute income levels. (We prefer health benefits more strongly at higher vs lower levels of income). Whether this shifts our portfolio more towards health- or income-generating interventions depends on our choice of income level at which we believe income doublings and health benefits should trade off as they currently do.