Consider those charities that expect their mission to take years rather than months. These charities will rationally want to spread their spending out over time. Particularly for charities with large endowments, they will attempt to use the interest on their money rather than depleting the principal, although if they expect to receive more donations over time they can be more liberal.

This means that a single donation slightly increases the rate at which such a charity does good, rather than enabling it to do things which it could not otherwise do. So the scaling factor of the endowment is restored: donating $1000 to a charity with a $10m endowment increases the rate at which it can sustainably spend by 1000/10^7 = 0.1%.

This does not mean that a charity will say, look, if our sustainable spending rate was 0.1% higher we'd have enough available this year to fund the 'save a million kids from starvation' project, oh well. They'll save the million kids and spend a bit less next year, all other things being equal. In other words, the charity, by maximising the good it does with the money it has, smooths out the change in its utility for small differences in spending relative to the size of its endowment, i.e. the higher order derivatives are low. So long as the utility you get from a charity comes from it fulfilling its stated mission, your utility will also vary smoothly with small spending differences.

Likewise, with rational collaborating charities, they will each adjust their spending to increase any mutually beneficial effects. So mixed derivatives are low, too.

The upshot is that unless your donation is of a size that it can permanently and significantly raise the spending power of such a charity, you won't be leaving the approximately linear neighbourhood in utility-space. So if you're looking for counterexamples, you'll need to find one of:

• charities with both low endowments and low donation rates, which nevertheless can produce massive positive effects with a smallish amount of money
• charities which must fulfil their mission in a short time and are just short of having the money to do so.