Calculating how much small donors funge with money that will never be spent

Epistemic status: Confident that the effect is real, though likely smaller than suggested by the toy-model.

Summary

Small donors should discount the cost effectiveness of their donations to interventions above a large funder’s bar if

1. they expect the large funder not to have spent all their capital by the time of AGI’s arrival^[1][2]
2. their donation to interventions above the large funder’s bar funges with the large funder.

In this post I describe a toy model to calculate how much to discount due to this effect.

I apply the model to a guess of Open Philanthropy’s  spending on Global Health and Development (GHD) with Metaculus’ AGI timelines (25% by 2029, 50% by 2039).  The model implies that small donors should consider interventions above OP's GHD bar, e.g. GiveWell's top charities, are only 55% as cost effective as the small donors first thought.  For shorter AGI timelines (25% by 2027, 50% by 2030) this factor is around 35%.

I use OP's GHD spending as an example because of their clarity around funding rate and bar for interventions. This discount factor would be larger if one funges with 'patient' philanthropic funds (such as The Patient Philanthropy Fund).

This effect is a corollary of the result that most donor's AGI timelines (e.g. deferral to Metaculus) imply that the community spend at a greater rate. When a small donor funges with a large donor (and saves them spending themselves),  the community's spending rate is effectively lowered (compared to when the small donor does not funge). 

This effect occurs when a small donor has shorter timelines than a large funder, or the large funder is not spending at a sufficiently high rate. In the latter case, small donors - by donating to interventions below the large funder's bar - are effectively correcting the community's implicit bar for funding.

Toy model

Suppose you have the choice of donating to one of two interventions, $A$ which gives $a$ utils per $, or $B$, which gives $b$ utils per $. Suppose further that the available interventions remain the same every year and that both have room for funding this year.

A large funder $F$ will ensure that $A$ is fully funded this year, so if you donate $1 to $A$, then $F$, effectively, has $1 more to donate in the future. I suppose that $F$ only ever donates to (opportunities as good as) $A$.

I suppose that $F$'s capital decreases by some constant amount $f$ times their initial capital each year.^[3] This means that $F$ will have no assets in $1/f$ years from now.

Supposing AGI arrives $t$ years from now, then F will have spent fraction $\text{min}(ft,1)$ of their current capital on $A$.

Accounting for this funging and assuming AGI arrives at time $t$, the cost effectiveness of your donation to $A$ is then $\text{min}(ft,1)\cdot a$ utils per $. Then if $b>\text{min}(ft,1)\cdot a$, marginal spending by small donors on $B$ is more cost effective than on $A$.

By considering distributions of AGI's arrival time $t$ and the large funder's funding rate $f$ we can get a distribution of this multiplier. 

Plugging in numbers

I take

• The large funder $F$ to be Open Philanthropy’s Global Health and Wellbeing spending on Global Health and Development and intervention $A$ to be Givewell’s recommendations. 
• I take $1/f$, the expected time until OP's funds dedicated to GHD are depleted to be distributed $Normal(20,20)$ bounded below by 5.^[4]
• I take AGI timelines to be an approximation those on Metaculus.

These distributions on AGI timelines and $1/f$ give the following distribution of the funging multiplier (reproducible here).

The ratio of cost effectiveness between GiveWell's recommendations and GiveDirectly, $a/b$, is approximately 7-8 ^[5]and so small donors should give  to interventions in the (5, 7)x GiveDirectly range. 

For donors with shorter timelines AGI timelines (25% by 2027, 50% by 3030) one should discount more.

Thanks to Tom Barnes for comments.

1. ^{^}

   And you believe that there is no use for capital post-AGI.

2. ^{^}

   One could also compute the analysis for any (existential) catastrophe that either affects the funder's ability to donate or the existence of interventions to give to.

3. ^{^}

   An earlier version of the model considered appreciation of capital and constant spending on $A$, though I have cut for simplicity to avoid discussion of patient 

   One could, for example, suppose that their capital is measured in "ability to do good" and that the interventions become more expensive as quickly as their capital appreciates. 

^{^}

This is some data and a lot of intuition.

In July 2022 Open Philanthropy wrote

Our current best estimate is that, for interventions like GiveWell’s recommended charities, our optimal strategy is to spend 8-10% of relevant assets each year.. This means that whatever level of assets we have, our cost-effectiveness “bar” for GiveWell-like interventions should be set so that the opportunities above this bar in the next year add up to 8-10% of such assets.

My distribution on $f$ implies they have roughly 20% change of meeting or exceeding this target. 

Eyeballing some historic spending and guesses of Dustin Moskovitz's wealth pus their historic spending rate at around 2-5%.

The lower bound $1/f>5$ is based on my current impression that OP's GHD team do not plan to spend all their funds within the next 5 years. A distribution other than trunctated normal would likely be more appropriate.

^{^}

Open Philanthropy, November 2021 

Overall, we currently expect GiveWell’s marginal cost-effectiveness to end up around 7-8x GiveDirectly (in their units),

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