Max, thanks for the post!
For someone like GiveWell that spends a lot of time investigating charities, they may have enough information about the charity’s budget to tell when there is (something similar to) a discrete jump in the derivative of the returns function. E.g. the way they talk about “capacity-relevant funding” and “execution funding” in the post you linked to (“incentive funding” is for a completely different purpose that has no direct relationship with returns).
Also, to fix ideas it helps to think what we represent by the funding axis on the impact against funding graph, i.e. returns function. Is the function specifying the relationship between total impact, and total funding the charity expects to receive for a time period (i.e. next year), or we are looking within a time period and plotting what the charity does as (unexpected) new money comes in? In the latter case, diminishing returns seems most likely. In the former case, increasing returns is possible (but diminishing returns is as well).
Ben Todd has written about increasing returns in small organizations here. I wrote here that “Whether returns are increasing or decreasing in additional funding depends on how the funding is received. Expecting a large chunk of funding (either in the form of receiving such amounts at once, or even expecting a total large amount received in small chunks if there is no lumpy investment or borrowing constraint) could enable an organization to do more risk taking, while getting unanticipated small amounts of funding at a time—even if the total adds up to more—will probably just lead the organization to use the marginal dollar to “fund the activity with the lowest (estimated) cost-effectiveness”. … The scenario Ben Todd has in mind probably applies more when a large funder is considering how much to give to an organization. This may be another argument to enter donor lottery or donate through the EA fund: giving a large and certain amount of donations to a small organization enables them to plan ahead for more risky but growth enhancing strategies, hence could be more valuable than uncoordinated small amounts even if the latter add up to the same total (because the latter may be less certain). … This mechanism is articulated in “5.2 The funding uncertainty problem” on this page about the EA fund.” (There are probably some analogous economic model of firm investment under liquidity constraint and uncertainty, but I don’t have one on the top of my head.)
In practice it may not be a big deal: even if the charity receives random small amounts of money during the year, it is probably at least as good as receiving the total amount all at once at the beginning of next year when they do the next round of planning. But for small organisations where earlier growth is much better, it could be much more preferable to have small amounts of donations be coordinated and committed at the same time to help with more ambitious planning and growth. (Of course we are assuming the charity is borrowing constrained; otherwise if earlier growth is much better they’d borrow to achieve it and repay with later donation. Also, if the market is efficient and earlier growth is really much better, then some donors should capture the opportunity … but of course market may not be!)
In response to your first paragraph, I think it’s true that GiveWell will have more information about any changes in the returns function. For the reasons given the in the second post, I think it’s unlikely that GiveWell charities do have inflection points in their returns functions. I’m not sure from GiveWell’s writing whether they think that there are inflection points or not (In particular, I don’t think they take a clear stance on this in the linked post).
I think your second paragraph is answered by footnote 1 of the first post. I don’t fully understand how your third and fourth paragraphs relate to the posts. Are you simply arguing that a fuller analysis would incorporate the size of individual donations, not the total level of funding? This seems like a plausible extension.
On increasing and decreasing (marginal) returns:
I see that you said “claiming that expected returns are normally diminishing is compatible with expecting that true returns increase over some intervals. I think that true returns often do increase over some intervals, but that returns generally decrease in expectation.”
I wasn’t sure why this would be true in a model that describes the organization’s behavior, so I spent some time thinking it through. Here is a way to reconcile increasing returns and decreasing expected returns, with a graph. Note that when talking about “funding” here (and the x-axis of the graph) I mean “funding the organization will receive over the next planning period, i.e. calendar year”, and assume there’s no uncertainty over funding received, same as in Max’s model.
I think it’s reasonable to assume that “increasing returns” in organization’s impact often come from cases of “lumpy investments”, i.e. things with high impact and high fixed costs. In this case nothing would happen until a certain level of funding is reached, and at that point there is a discrete jump in impact. For the sake of the argument let’s assume that everything the organization does has this nature (we’ll relax this later). So you’d expect the true returns function to be a step function (see the black curve on graph).
How does the organization makes decision? First, let’s assume that these “lumpy investments” (call them “projects”) aren’t actually 0 or 1; rather, the closer the level of funding is to the “required” level, the more likely the project will happen (e.g. maybe AMF is trying to place an order for bed nets and the minimum requirement is 1000 nets, but it’s possible that they can convince the supplier to make an order of 900 nets with probability less than 1). For simplicity let’s assume the probability grows linearly (we’ll relax this later). Then the expected returns is actually the red piecewise linear function in the graph. Note that overall the marginal returns are still weakly diminishing (but they are constant within each project) because given the red expected returns function the organization would choose to first do the project with the highest marginal return (i.e. slope), then the second highest, etc.
Note: We assume the probability grows linearly. If we relax this assumption, things get more complicated. I illustrate the case where probabilities grow in a convex way within each project with the ugly green curves (note that this also covers the case with no uncertainty in the project happening or not, but rather the project has a “continuous” nature and increasing marginal returns). It’s true that you cannot call the whole thing concave (and I don’t know if mathematicians have a word to describe something like this). But from the perspective of a donor who, IN ADDITION to the model here that assumes certainty in funding levels, has uncertainty over how much funding the organization has, the “expected-expected” returns function they face (with expectation over funding level and impact) would probably be closer to the earlier piecewise linear thing, or concave. If the probabilities grow in some weird ways that are not completely convex (note that this also covers the case with no uncertainty in the project happening or not, but rather the project has a “continuous” nature and weirdly shaped, non-convex marginal returns), things may get more complicated (e.g. switching projects half way may happen if the organization always spends the next dollar on the next thing with highest marginal return) -- maybe we should abandon such possibilities since they are unintuitive.
Note: If the organization does some projects that look more like linear in the relationship between impact and funding, 1) we can still use the red piecewise linear graph, and organizations will still start with projects with the highest slopes; 2) at a fine level things are still discrete so we’ll be back to (mini) step functions.
Note: We also assumed the only uncertainty here is whether a project would happen at a funding level less than “required”. There could also be uncertainty over impact, conditional the project happening—this is not in our model, but my guess is it shouldn’t change the main results much (of course it might depend on the shape of the new layer of uncertainty, and I haven’t thought about it carefully).
All of the above is essentially based on the old idea that organizations do highest returns things first. The main addition is to look at a model where there are discrete projects (with elements of increasing returns) and still arrive at the same general conclusion.
I don’t know how many people find this useful, but I was very confused by this issue (and said some incoherent things in my earlier comments, which I’ve delete to avoid confusing people), and found that I had to think through what the organization actually does in the case of lumpy investments.
Other important issues that are related but out of the scope of this discussion include how organizations and donors act under uncertainty over donation to be received by the organization.