An under-appreciated observation about giving now vs later

"...these incredible rates of compounding must decline over time. Otherwise these small charitable acts would quickly become responsible for all of the good in the world. What sustainable level of compounding can they achieve in the long run? The global economic growth rate..."

"High-risk business investments by nature compound more quickly than the global economic growth rate. This is the classic 'r > g', which has become widely understood due to Piketty."

Maybe I'm missing something, because I see this pair argued a lot, but when phrased this way this pair of arguments seem to be in tension.

I understand the argument that donations earning even 2% above the global economic growth rate in the very long run would result in a single small donation eventually being responsible for virtually all good done (a seemingly absurd conclusion), but by the exact same token that means a high-risk business investment earning even 2% above the global economic growth rate in the very long run would eventually be making annual returns greater than annual world GDP (an equally absurd conclusion). So in fact both must eventually decay to the world growth rate.

It must be said that Paul, AFAIK, doesn't frame the argument this way. From the post you linked to:

"Here’s the problem: if you give your money well, it might compound much faster than it would have in your bank account—but only for a while. Over time the positive effects will spread out more and more, across a broader and broader group of beneficiaries, until eventually you are just contributing to a representative basket of all human activities. Eventually, my investment will compound at the rate of world economic growth, rather than at the particularly promising interest rate I was able to originally identify."

In other words, with an investment the return keeps coming in to you and you can continue to direct it to activites which are above-average promising (or above-average risky). But with a donation you essentially lose control over it immediately and so you shouldn't expect it to stay in above-average promising territory for very long at all; the timeline on 'eventually' is short. If you could somehow capture a fraction that initial high social return and maintain control of it, e.g. if from that initial 20% return on cash transfers you received 6%, then donating would dominate, but in practice that's very difficult. Microfinance is arguably an attempt to do exactly this, but unfortunately doesn't appear to have the great social returns we're looking for.

Off-topic, but thinking about it in terms of control also highlights why meta-activities might well be in a different category. If you give money to a meta-activity that generates 'EA dollars' at a high enough multiplier on a short enough timeframe, then while you've lost control of your money the control has been passed on to people who you might reasonably expect to also be searching for above-average promising activities.

More detailed models and arguments along the same lines here: https://80000hours.org/articles/should-you-wait/

For the same of illustration, let's say the global economic growth rate is 3%, the return on business equity is 5%, and the internal rate of compounding of, say, cash transfers is 20% in the first year, 18% in the second, then 16%, 14%, 12%, 10%, 8%, 6%, 4%, and then 3% thereafter. Holding all else constant, how long would the 'invest and give later' strategy take to catch up?...40 years

It looks like you're comparing giving now to investing (but never giving), not investing and giving later. The latter requires specifying when the 'giving later' happens.

With your figures you get 8 years before donation returns fall below business equity's 5% rate. So saving for one year, then donating, will come out ahead of donating now on a 9 year time scale, not a 40 year time scale.

So the graph and '40 years' don't match up with 'invest and give later' rather than 'invest and give never.'