I don’t have a good object-level answer, but maybe thinking through this model can be helpful.
Big picture description: We think that a person’s impact is heavy tailed. Suppose that the distribution of a person’s impact is determined by some concave function of hours worked. We want that working more hours increases the mean of the impact distribution, and probably also the variance, given that this distribution is heavy-tailed. But we plausibly want that additional hours affect the distribution less and less, if we’re prioritising perfectly (as Lukas suggests) -- that’s what concavity gives us. If talent and luck play important roles in determining impact, then this function will be (close to) flat, so that additional hours don’t change the distribution much. If talent is important, then the distributions for different people might be quite different and signals about how talented a person is are informative about what their distribution looks like.
This defines a person’s expected impact in terms of hours worked. We can then see whether this function is linear or concave or convex etc., which will answer your question.
More concretely: suppose that a person’s impact is lognormally distributed with parameters μ and σ, that μ is an increasing, concave function of hours worked, h, and that σ is fixed. I chose this formulation because it’s simple but still enlightening, and has some important features: expected impact, eμ(h)+σ22, is increasing in hours worked and the variance is also increasing in hours worked. I’m leaving σ fixed for simplicity. Suppose also that μ(h)=logh, which then implies that expected impact is heσ22, i.e. expected impact is linear in hours worked.
Obviously, this probably doesn’t describe reality very well, but we can ask what changes if we change the underlying assumptions. For example, it seems pretty plausible that impact is heavier-tailed than lognormally distributed, which suggests, holding everything else equal, that expected impact is convex in hours worked, so you lose more than 20% impact by working 20% less.
Getting a good sense of what the function of hours worked (here μ(h)) should look like is super hard in the abstract, but seems more doable in concrete cases like the one described above. Here, the median impact is eμ(h)=h, if μ(h)=logh, so the median impact is linear in hours worked. This doesn’t seem super plausible to me. I’d guess that the median impact is concave in hours worked, which would require μ to be more concave than log, which suggests, holding everything else equal, that expected impact is concave in hours worked. I’m not sure how this changes if you consider other distributions though—it’s a peculiarity of the lognormal distribution that the mean is linear in the median, if σ is held fixed, so things could look quite different with other distributions (or if we tried to determine μ and σ from h jointly).
Median impact being linear in hours worked seems unlikely globally—like, if I halved my hours, I think I’d more than half my median impact; if I doubled them, I don’t think I would double my median impact (setting burnout concerns aside). But it seems more plausible that median impact could be close to linear over the margins you’re talking about. So maybe this suggests that the model isn’t too bad for median impact, and that if impact is heavier-tailed than lognormal, then expected impact is indeed convex in hours worked.
This doesn’t directly answer your question very well but I think you could get a pretty good intuition for things by playing around with a few models like this.
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!
I don’t have a good object-level answer, but maybe thinking through this model can be helpful.
Big picture description: We think that a person’s impact is heavy tailed. Suppose that the distribution of a person’s impact is determined by some concave function of hours worked. We want that working more hours increases the mean of the impact distribution, and probably also the variance, given that this distribution is heavy-tailed. But we plausibly want that additional hours affect the distribution less and less, if we’re prioritising perfectly (as Lukas suggests) -- that’s what concavity gives us. If talent and luck play important roles in determining impact, then this function will be (close to) flat, so that additional hours don’t change the distribution much. If talent is important, then the distributions for different people might be quite different and signals about how talented a person is are informative about what their distribution looks like.
This defines a person’s expected impact in terms of hours worked. We can then see whether this function is linear or concave or convex etc., which will answer your question.
More concretely: suppose that a person’s impact is lognormally distributed with parameters μ and σ, that μ is an increasing, concave function of hours worked, h, and that σ is fixed. I chose this formulation because it’s simple but still enlightening, and has some important features: expected impact, eμ(h)+σ22, is increasing in hours worked and the variance is also increasing in hours worked. I’m leaving σ fixed for simplicity. Suppose also that μ(h)=logh, which then implies that expected impact is heσ22, i.e. expected impact is linear in hours worked.
Obviously, this probably doesn’t describe reality very well, but we can ask what changes if we change the underlying assumptions. For example, it seems pretty plausible that impact is heavier-tailed than lognormally distributed, which suggests, holding everything else equal, that expected impact is convex in hours worked, so you lose more than 20% impact by working 20% less.
Getting a good sense of what the function of hours worked (here μ(h)) should look like is super hard in the abstract, but seems more doable in concrete cases like the one described above. Here, the median impact is eμ(h)=h, if μ(h)=logh, so the median impact is linear in hours worked. This doesn’t seem super plausible to me. I’d guess that the median impact is concave in hours worked, which would require μ to be more concave than log, which suggests, holding everything else equal, that expected impact is concave in hours worked. I’m not sure how this changes if you consider other distributions though—it’s a peculiarity of the lognormal distribution that the mean is linear in the median, if σ is held fixed, so things could look quite different with other distributions (or if we tried to determine μ and σ from h jointly).
Median impact being linear in hours worked seems unlikely globally—like, if I halved my hours, I think I’d more than half my median impact; if I doubled them, I don’t think I would double my median impact (setting burnout concerns aside). But it seems more plausible that median impact could be close to linear over the margins you’re talking about. So maybe this suggests that the model isn’t too bad for median impact, and that if impact is heavier-tailed than lognormal, then expected impact is indeed convex in hours worked.
This doesn’t directly answer your question very well but I think you could get a pretty good intuition for things by playing around with a few models like this.
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!
Thanks Aidan, I’ll consider this model when doing any more thinking on this.