This post has been sitting in my open tabs for 4 months and I am finally getting to it today.
Using η=1.87 would require us to start taking into account absolute income levels in our cost-effectiveness analyses.
I’m not entirely sure, but by my reading, the article is using this as an argument against η > 1. But I don’t think it’s really an argument. If η > 1 then indeed you should take into account absolute income levels, and I do in fact think you should do that. And yes that would change prioritization, and that’s a good thing because the current prioritization is probably wrong—it doesn’t assign enough weight to people with lower incomes.
You never actually say you’re arguing against η=1.87, but the title implies it.
FWIW I am not convinced by the evidence on what value of η to use, different lines of evidence point in different directions. I lean toward η > 1, so I think it’s reasonable to use η > 1 everywhere.
I think most people intuitively feel that income doublings matter more for poorer people, which requires η > 1.
η = 1 implies you can get arbitrarily high utility if you’re sufficiently rich, which seems wrong. It seems more likely that income doublings provide diminishing returns—once you reach very high income levels (like $100M+), income doublings hardly matter at all. This makes sense when you look at the space of consumable goods: doubling your income increases the size of the set of things you can buy, but each doubling increases the set size by less than the previous doubling.[1]
The best literature review I’ve seen comes from Gordon Irlam. The only real conclusion is that different methods see huge variance in estimates of η.
One paper that I particularly like[2], A New Method of Estimating Risk Aversion, estimates η using labor elasticity and finds η = 0.71 (see Table 1). But if you look at the various data sources it uses, the estimates of η vary greatly depending on source so the η = 0.71 average hides a lot of underlying uncertainty.
I often see people cite life satisfaction data showing η = 1. The commonly-cited paper, Stevenson & Wolfers (2013), didn’t perform any statistical tests for non-linearity on log-income vs. happiness. In Table 1, the paper did binary comparisons of the slope of log-income vs. happiness for rich vs. poor people and did not find clear differences, but it did find that the slope was generally steeper for rich people[3], which suggests η < 1 (I’m pretty sure on priors that η >= 1 so I don’t know what’s up with that result).
I briefly looked for more recent papers that test for non-linearity of log-income vs. happiness. I didn’t find exactly that, but I did find Happiness, income satiation and turning points
around the world which finds that life satisfaction levels off at a certain income level. I didn’t read carefully but it looks like this paper used a sketchy spline curve-fitting method that I don’t trust (the fitted curves show that higher income decreases happiness above a certain point, which suggests that they’re using the wrong kind of curve; see Fig 1 and Fig 2[4]). But the fact that their spline curves level off suggests that happiness increases sub-linearly with income.
I feel like there’s room for a solid meta-analysis on income and life satisfaction, and I’m not satisfied with any of the existing literature.
In summary, the existing evidence is so high-variance that none of it meaningfully updates me away from my intuition that η must be greater than 1.
[1] This brings to mind a method for estimating η that I’ve never seen: Assume the prices of goods are Pareto-distributed and estimate the alpha parameter of the underlying Pareto distribution. Use that to estimate η (using this method).
[2] Even though I haven’t actually read most of it lol. I just like the concept. Maybe it contains a bunch of math errors, I don’t know.
[3] The paper did this comparison across a bunch of surveys. You’d need to do some kind of sophisticated non-standard significance test to determine if the overall difference is statistically significant, and the paper did not do that. (I think what you’d want to do is create a combined likelihood function that includes every survey and then get the p-value from the likelihood function. Or just skip the p-value and report the shape of the likelihood function because that’s more informative anyway.)
[4] Perhaps this is a property of the data, not the curve-fitting method. The paper says “in [some comparisons], the SWB [subjective well-being] level at satiation was greater due to turning-point effects (Bayes factor < 1⁄3).” They say they present this data in the supplementary appendix, which isn’t publicly available and isn’t on Sci-Hub (AFAICT), so it seems I can’t check.
This post has been sitting in my open tabs for 4 months and I am finally getting to it today.
I’m not entirely sure, but by my reading, the article is using this as an argument against η > 1. But I don’t think it’s really an argument. If η > 1 then indeed you should take into account absolute income levels, and I do in fact think you should do that. And yes that would change prioritization, and that’s a good thing because the current prioritization is probably wrong—it doesn’t assign enough weight to people with lower incomes.
You never actually say you’re arguing against η=1.87, but the title implies it.
FWIW I am not convinced by the evidence on what value of η to use, different lines of evidence point in different directions. I lean toward η > 1, so I think it’s reasonable to use η > 1 everywhere.
I think most people intuitively feel that income doublings matter more for poorer people, which requires η > 1.
η = 1 implies you can get arbitrarily high utility if you’re sufficiently rich, which seems wrong. It seems more likely that income doublings provide diminishing returns—once you reach very high income levels (like $100M+), income doublings hardly matter at all. This makes sense when you look at the space of consumable goods: doubling your income increases the size of the set of things you can buy, but each doubling increases the set size by less than the previous doubling.[1]
The best literature review I’ve seen comes from Gordon Irlam. The only real conclusion is that different methods see huge variance in estimates of η.
One paper that I particularly like[2], A New Method of Estimating Risk Aversion, estimates η using labor elasticity and finds η = 0.71 (see Table 1). But if you look at the various data sources it uses, the estimates of η vary greatly depending on source so the η = 0.71 average hides a lot of underlying uncertainty.
I often see people cite life satisfaction data showing η = 1. The commonly-cited paper, Stevenson & Wolfers (2013), didn’t perform any statistical tests for non-linearity on log-income vs. happiness. In Table 1, the paper did binary comparisons of the slope of log-income vs. happiness for rich vs. poor people and did not find clear differences, but it did find that the slope was generally steeper for rich people[3], which suggests η < 1 (I’m pretty sure on priors that η >= 1 so I don’t know what’s up with that result).
I briefly looked for more recent papers that test for non-linearity of log-income vs. happiness. I didn’t find exactly that, but I did find Happiness, income satiation and turning points around the world which finds that life satisfaction levels off at a certain income level. I didn’t read carefully but it looks like this paper used a sketchy spline curve-fitting method that I don’t trust (the fitted curves show that higher income decreases happiness above a certain point, which suggests that they’re using the wrong kind of curve; see Fig 1 and Fig 2[4]). But the fact that their spline curves level off suggests that happiness increases sub-linearly with income.
I feel like there’s room for a solid meta-analysis on income and life satisfaction, and I’m not satisfied with any of the existing literature.
In summary, the existing evidence is so high-variance that none of it meaningfully updates me away from my intuition that η must be greater than 1.
[1] This brings to mind a method for estimating η that I’ve never seen: Assume the prices of goods are Pareto-distributed and estimate the alpha parameter of the underlying Pareto distribution. Use that to estimate η (using this method).
[2] Even though I haven’t actually read most of it lol. I just like the concept. Maybe it contains a bunch of math errors, I don’t know.
[3] The paper did this comparison across a bunch of surveys. You’d need to do some kind of sophisticated non-standard significance test to determine if the overall difference is statistically significant, and the paper did not do that. (I think what you’d want to do is create a combined likelihood function that includes every survey and then get the p-value from the likelihood function. Or just skip the p-value and report the shape of the likelihood function because that’s more informative anyway.)
[4] Perhaps this is a property of the data, not the curve-fitting method. The paper says “in [some comparisons], the SWB [subjective well-being] level at satiation was greater due to turning-point effects (Bayes factor < 1⁄3).” They say they present this data in the supplementary appendix, which isn’t publicly available and isn’t on Sci-Hub (AFAICT), so it seems I can’t check.