Thanks for this! I think the setup is excellent, especially the diagram that makes it very clear what’s going on. It seems basically comprehensive to me—not fully comprehensive, but it covers the most important stuff.
My approach when reading the full paper was:
What are the interesting conclusions?
What model assumptions produce those conclusions?
Are any assumptions worth changing, and how might that change the conclusion?
The main conclusions, as I see it:
Labor grows to a constant size, while capital keeps growing.
Fraction of labor dedicated to earning-to-give approaches 0.
I like most of the assumptions, and I like that you can get interesting conclusions without knowing much about the parameter values.
I noticed three main things that I didn’t see addressed (much) in the paper. From most to least important:
Model assumes that labor experiences depreciation, while capital does not. But that’s not true. Individuals who control capital might leave the movement; nonprofits that control capital might experience value drift. Prima facie, labor and capital should experience the same rate of depreciation[1]. The assumption of non-depreciating capital produced the results that (a) capital grows faster than movement size and (b) asymptotically 100% of laborers are hired from outside the movement. My intuition is that if you adjust this assumption, the optimal allocation will have both capital and labor asymptoting to a fixed size, and kDt=ℓDt and kRt=ℓRt, with kRt,ℓRt asymptoting to 0. I’m less confident about this, but I’d guess that the optimal fraction of labor spent on earning-to-give would asymptotically equal the fraction of capital spent on recruiting, so that we’re investing the same amount in both labor and capital.
This seems important because the assumption of depreciating capital is more realistic, and incorporating it would (maybe?) change both of the main conclusions.
If capital depreciates, but at a slower rate than labor, my intuition is that the optimal fraction of earning-to-give still approaches 0. If we model the movement as having “committed” members and “casual” members with different attrition rates, and with capital disproportionately coming from committed members, I’d guess you would find that (a) both labor and capital come almost exclusively from committed members in the long run, and (b) optimal earning-to-give fraction is a positive constant because committed labor and committed capital depreciate at the same rate.
Those adjusted conclusions are based on my intuitions. I’m not good enough at dynamic optimization to actually derive the solutions.
The paper briefly addresses this, but it’s possible to recruit capital as well as labor by finding wealthy donors. (This is kind of the inverse of my first point.) Recruitment is zero-sum, so it can’t be true in general that recruiting capital is easier than recruiting labor, but historically it looks like EAs have an easier time recruiting capital.
A patient movement cannot achieve market return indefinitely. Market return equals price change plus yield, and in the aggregate, investors do not re-invest their yield (I think?[2]). But patient investors always re-invest a positive portion of their yield, so their fraction of global wealth will increase over time. Eventually, this will decrease the rate of return on capital. This will take a long time (and might never happen in real life), but it matters when you’re looking at asymptotic behavior.
A final note: I find it hard to read mathy papers that use a bunch of single-letter variables that are defined once in the middle of a paragraph, because I forget the definitions and then have trouble finding them again. It would be very helpful if the paper included a table of variable definitions.
[1] I would guess that, in practice, capital depreciates more slowly because capital is disproportionately controlled by the most committed members of the movement.
[2] Here is why I think this is true: For stocks, at least, the dividend yield comes from company earnings. In the long run, if the yield is constant, that means stock price growth equals earnings growth. If investors re-invest their dividends, which will push price growth above earnings growth, decreasing yield.
It’s also possible that yield could decrease over time, but that gives the same conclusion—namely, that return on capital decreases.
Although I’m not sure that’s how prices work, eg maybe stock prices are highly inelastic, so re-investing dividend yields doesn’t cause them to go up significantly.
Re 1: if I’m understanding you right, this would just lower the interest rate from r to r—capital ‘depreciation rate’. So it wouldn’t change any of the qualitative conclusions, except that it would make it more plausible that the EA movement (or any particular movement) is, for modeling purposes, “impatient”. But cool, that’s an important point. And particularly relevant these days; my understanding is that a lot of Will’s(/etc) excitement around finding megaprojects ASAP is driven by the sense that if we don’t, some of the money will wander off.
Re 2: another good point. In this case I just think it would make the big qualitative conclusion hold even more strongly—no need to earn to give because money is even easier to come by, relative to labor, than the model suggests. But maybe it would be worth working through it after adding an explicit “wealth recruitment” function, to make sure there are no surprises.
Re 3: I agree, but I suspect—perhaps pessimistically—that the asymptotics of this model (if it’s roughly accurate at all) bite a long time before EA wealth is a large enough fraction of global capital to push down the interest rate! Indeed, I don’t think it’s crazy to think they’re already biting. Presumably the thing to do if you actually got to that point would be to start allocating more resources to R&D, to raise labor productivity and thus the return to capital. There are many ways I’d want to make the model more realistic before worrying about the constraints you run into when you start owning continents (a scenario for which there would presumably be plenty of time to prepare...!); but as noted, one of the extensions I’m hoping gets done before too long is to make (at least certain kinds of) R&D endogenous. So hopefully that would be at least somewhat relevant.
Hey, in hindsight I realize that the paper + summarization don’t make clear that this does depend on model assumptions/empirical points, sorry. I’ve edited the post to make this clearer (here is the previous version without the edits, in case it’s of interest.)
tl;dr: This comes from model assumptions which seem reasonable, but empirical investigations + historical case studies, or alternatively sci-fi scenarios could flip the conclusion.
In particular, let L′=−r⋅L+f(a⋅L,b⋅K), i.e. roughly L(t)=L(t−1)⋅(1−r)+f(a⋅L,b⋅K), so each year you lose r% of people, but you also do some movement building, for which you spend a⋅L labor and b⋅K capital.
Then for some functions f which determine movement building, this already implies that the movement has a maximum size. So for instance, if you havef(a⋅L,b⋅K)=log(11a⋅L+1b⋅K), then with infinite capital this reduces to f(a⋅L,b⋅K)=log(11a⋅L+1∞)=log(11a⋅L)=log(a⋅L)
But then even if you allocate all labor to movement building (so that a=1, or something), you’d have something like L′=−r⋅L+log(L) , and this eventually converges to the point where log(L)=r⋅L no matter where you start.
Now, above I’ve omitted some constants, and our function isn’t quite the same, but that’s essentially what’s going on (see ρR<0,λ<1 in equation 6 in page 4.) I.e., if you lose movement participants as a percentage but have a recruitment function that eventually has “brutal” diminishing returns (sub-linear diminishing returns to labor + throwing money at movement building doesn’t solve it), you get a similar result (movement converges to a constant.)
But you could also imagine a scenario where the returns are less brutal—e.g., you’re always able to recruit an additional participant by throwing money at the problem, or every movement builder can sort of eternally always recruit a person every year, etc. You could also imagine more sci-fi like scenario, where humanity is expanding exponentially (cubically) in space, and a social movement is a constant fraction of humanity.
More realistically, if f instead looks like √(a∗L)⋅(b∗K), which has diminishing returns but not brutally so, movement size can increase forever because you can always throw more money at the problem until √(a∗L)⋅(b∗K)>r⋅L
Note that if you have a less brutal recruitment function, this increases the appeal of movement building, not of earn to giving.
Also, I’m not sure whether “brutal” is the right way to be talking about this. “Brutal” is the term I use when I think about this but if I recall correctly the function we use is standard in the literature, and it seems plausible when you start to think about groups which reach a large size. But there is definitely an empirical question here about how movement results actually look like.
Hey, in hindsight I realize that the paper + summarization don’t make clear that this does depend on model assumptions/empirical points
FWIW this was clear to me, I was using “conclusions” to mean “conclusions, given the model assumptions”, not “conclusions, which the authors definitely think are true”.
Thanks for this! I think the setup is excellent, especially the diagram that makes it very clear what’s going on. It seems basically comprehensive to me—not fully comprehensive, but it covers the most important stuff.
My approach when reading the full paper was:
What are the interesting conclusions?
What model assumptions produce those conclusions?
Are any assumptions worth changing, and how might that change the conclusion?
The main conclusions, as I see it:
Labor grows to a constant size, while capital keeps growing.
Fraction of labor dedicated to earning-to-give approaches 0.
I like most of the assumptions, and I like that you can get interesting conclusions without knowing much about the parameter values.
I noticed three main things that I didn’t see addressed (much) in the paper. From most to least important:
Model assumes that labor experiences depreciation, while capital does not. But that’s not true. Individuals who control capital might leave the movement; nonprofits that control capital might experience value drift. Prima facie, labor and capital should experience the same rate of depreciation[1]. The assumption of non-depreciating capital produced the results that (a) capital grows faster than movement size and (b) asymptotically 100% of laborers are hired from outside the movement. My intuition is that if you adjust this assumption, the optimal allocation will have both capital and labor asymptoting to a fixed size, and kDt=ℓDt and kRt=ℓRt, with kRt,ℓRt asymptoting to 0. I’m less confident about this, but I’d guess that the optimal fraction of labor spent on earning-to-give would asymptotically equal the fraction of capital spent on recruiting, so that we’re investing the same amount in both labor and capital.
This seems important because the assumption of depreciating capital is more realistic, and incorporating it would (maybe?) change both of the main conclusions.
If capital depreciates, but at a slower rate than labor, my intuition is that the optimal fraction of earning-to-give still approaches 0. If we model the movement as having “committed” members and “casual” members with different attrition rates, and with capital disproportionately coming from committed members, I’d guess you would find that (a) both labor and capital come almost exclusively from committed members in the long run, and (b) optimal earning-to-give fraction is a positive constant because committed labor and committed capital depreciate at the same rate.
Those adjusted conclusions are based on my intuitions. I’m not good enough at dynamic optimization to actually derive the solutions.
The paper briefly addresses this, but it’s possible to recruit capital as well as labor by finding wealthy donors. (This is kind of the inverse of my first point.) Recruitment is zero-sum, so it can’t be true in general that recruiting capital is easier than recruiting labor, but historically it looks like EAs have an easier time recruiting capital.
A patient movement cannot achieve market return indefinitely. Market return equals price change plus yield, and in the aggregate, investors do not re-invest their yield (I think?[2]). But patient investors always re-invest a positive portion of their yield, so their fraction of global wealth will increase over time. Eventually, this will decrease the rate of return on capital. This will take a long time (and might never happen in real life), but it matters when you’re looking at asymptotic behavior.
A final note: I find it hard to read mathy papers that use a bunch of single-letter variables that are defined once in the middle of a paragraph, because I forget the definitions and then have trouble finding them again. It would be very helpful if the paper included a table of variable definitions.
[1] I would guess that, in practice, capital depreciates more slowly because capital is disproportionately controlled by the most committed members of the movement.
[2] Here is why I think this is true: For stocks, at least, the dividend yield comes from company earnings. In the long run, if the yield is constant, that means stock price growth equals earnings growth. If investors re-invest their dividends, which will push price growth above earnings growth, decreasing yield.
It’s also possible that yield could decrease over time, but that gives the same conclusion—namely, that return on capital decreases.
Although I’m not sure that’s how prices work, eg maybe stock prices are highly inelastic, so re-investing dividend yields doesn’t cause them to go up significantly.
Thanks! A lot of good points here.
Re 1: if I’m understanding you right, this would just lower the interest rate from r to r—capital ‘depreciation rate’. So it wouldn’t change any of the qualitative conclusions, except that it would make it more plausible that the EA movement (or any particular movement) is, for modeling purposes, “impatient”. But cool, that’s an important point. And particularly relevant these days; my understanding is that a lot of Will’s(/etc) excitement around finding megaprojects ASAP is driven by the sense that if we don’t, some of the money will wander off.
Re 2: another good point. In this case I just think it would make the big qualitative conclusion hold even more strongly—no need to earn to give because money is even easier to come by, relative to labor, than the model suggests. But maybe it would be worth working through it after adding an explicit “wealth recruitment” function, to make sure there are no surprises.
Re 3: I agree, but I suspect—perhaps pessimistically—that the asymptotics of this model (if it’s roughly accurate at all) bite a long time before EA wealth is a large enough fraction of global capital to push down the interest rate! Indeed, I don’t think it’s crazy to think they’re already biting. Presumably the thing to do if you actually got to that point would be to start allocating more resources to R&D, to raise labor productivity and thus the return to capital. There are many ways I’d want to make the model more realistic before worrying about the constraints you run into when you start owning continents (a scenario for which there would presumably be plenty of time to prepare...!); but as noted, one of the extensions I’m hoping gets done before too long is to make (at least certain kinds of) R&D endogenous. So hopefully that would be at least somewhat relevant.
Re: Labor grows to a constant size
Hey, in hindsight I realize that the paper + summarization don’t make clear that this does depend on model assumptions/empirical points, sorry. I’ve edited the post to make this clearer (here is the previous version without the edits, in case it’s of interest.)
tl;dr: This comes from model assumptions which seem reasonable, but empirical investigations + historical case studies, or alternatively sci-fi scenarios could flip the conclusion.
In particular, let L′=−r⋅L+f(a⋅L,b⋅K), i.e. roughly L(t)=L(t−1)⋅(1−r)+f(a⋅L,b⋅K), so each year you lose r% of people, but you also do some movement building, for which you spend a⋅L labor and b⋅K capital.
Then for some functions f which determine movement building, this already implies that the movement has a maximum size. So for instance, if you havef(a⋅L,b⋅K)=log(11a⋅L+1b⋅K), then with infinite capital this reduces to f(a⋅L,b⋅K)=log(11a⋅L+1∞)=log(11a⋅L)=log(a⋅L)
But then even if you allocate all labor to movement building (so that a=1, or something), you’d have something like L′=−r⋅L+log(L) , and this eventually converges to the point where log(L)=r⋅L no matter where you start.
Now, above I’ve omitted some constants, and our function isn’t quite the same, but that’s essentially what’s going on (see ρR<0,λ<1 in equation 6 in page 4.) I.e., if you lose movement participants as a percentage but have a recruitment function that eventually has “brutal” diminishing returns (sub-linear diminishing returns to labor + throwing money at movement building doesn’t solve it), you get a similar result (movement converges to a constant.)
But you could also imagine a scenario where the returns are less brutal—e.g., you’re always able to recruit an additional participant by throwing money at the problem, or every movement builder can sort of eternally always recruit a person every year, etc. You could also imagine more sci-fi like scenario, where humanity is expanding exponentially (cubically) in space, and a social movement is a constant fraction of humanity.
More realistically, if f instead looks like √(a∗L)⋅(b∗K), which has diminishing returns but not brutally so, movement size can increase forever because you can always throw more money at the problem until √(a∗L)⋅(b∗K)>r⋅L
Note that if you have a less brutal recruitment function, this increases the appeal of movement building, not of earn to giving.
Also, I’m not sure whether “brutal” is the right way to be talking about this. “Brutal” is the term I use when I think about this but if I recall correctly the function we use is standard in the literature, and it seems plausible when you start to think about groups which reach a large size. But there is definitely an empirical question here about how movement results actually look like.
FWIW this was clear to me, I was using “conclusions” to mean “conclusions, given the model assumptions”, not “conclusions, which the authors definitely think are true”.
Right, thanks, it seemed better to be too paranoid than to be too little.