After a little more thought, I think it might be helpful to think about/look into the relationship between the mean and median of heavy-tailed distributions and in particular, whether the mean is ever exponential in the median.
I think we probably have a better sense of the relationship between hours worked and the median than between hours worked and the mean because the median describes “typical” outcomes and means are super unintuitive and hard to reason about for very heavy tailed distributions. In particular, arguments like those given by Hauke seem more applicable to the median than the mean. This suggests that the median is roughly logarithmic in hours worked. It would then require the mean to be exponential in the median for the mean to be linear in hours worked, in which case, working 20% less would lose exactly 20% of the expected impact (more if the mean is more convex than exponential in the median, less if it’s less than exponential).
In the simple example above, the mean is linear in the median, so the mean is logarithmic in hours worked if the median is. But the lognormal distribution might not be heavy-tailed enough, so I wouldn’t put too much weight on this.
Looking at the pareto distribution, it seems to be the case that the mean is sometimes more than exponential in the median—it’s less convex for small values and more convex for high values . You’d have to a bit of work to figure out the scale and whether it’s more than exponential over the relevant range, but it could turn out that expected impact is convex in hours worked in this model, which would suggest working 20% less would lose more than 20% of the value. I’m not sure how well the pareto distribution describes the median though (it seems good for heavy tails but bad for the whole distribution of things), so it might be better to look at something like a lognormal body with a pareto tail. But maybe that’s getting too complicated to be worth it. This seems like an interesting and important question though, so I might spend more time thinking about it!
After a little more thought, I think it might be helpful to think about/look into the relationship between the mean and median of heavy-tailed distributions and in particular, whether the mean is ever exponential in the median.
I think we probably have a better sense of the relationship between hours worked and the median than between hours worked and the mean because the median describes “typical” outcomes and means are super unintuitive and hard to reason about for very heavy tailed distributions. In particular, arguments like those given by Hauke seem more applicable to the median than the mean. This suggests that the median is roughly logarithmic in hours worked. It would then require the mean to be exponential in the median for the mean to be linear in hours worked, in which case, working 20% less would lose exactly 20% of the expected impact (more if the mean is more convex than exponential in the median, less if it’s less than exponential).
In the simple example above, the mean is linear in the median, so the mean is logarithmic in hours worked if the median is. But the lognormal distribution might not be heavy-tailed enough, so I wouldn’t put too much weight on this.
Looking at the pareto distribution, it seems to be the case that the mean is sometimes more than exponential in the median—it’s less convex for small values and more convex for high values . You’d have to a bit of work to figure out the scale and whether it’s more than exponential over the relevant range, but it could turn out that expected impact is convex in hours worked in this model, which would suggest working 20% less would lose more than 20% of the value. I’m not sure how well the pareto distribution describes the median though (it seems good for heavy tails but bad for the whole distribution of things), so it might be better to look at something like a lognormal body with a pareto tail. But maybe that’s getting too complicated to be worth it. This seems like an interesting and important question though, so I might spend more time thinking about it!