In general, diminishing returns. As an intervention is being funded and makes an impact, eventually getting additional impact on the margin is more and more expensive. As a general rule, you should always fund the most cost-effective intervention - which means funding each intervention up to the point it stops being the most cost-effective one, and then switching to new-best thing.
I think people tend to jump straight to debating whether we should pursue charity by cost-effectiveness. That seems unproductive to me unless people have some idea what those costs actually are. Advertising like this seriously undermines any hope of that.
We could imagine, and UNICEF might argue, that these ads just motivate people to give money and don't set their expectations long-term. But when this is actually surveyed, we find that the median person thinks you can save a life for <$50. That's concerning on two levels. First, it means any charity that honestly advertises how much it spends to help people will look terrible. Second, the worrying question of what people would do if they knew the real number. Would people slash their donations if they thought the money would go 1% as far? Increase them to do equal good? Or is the amount we give to charity just not meaningfully correlated with how much it helps?
I wish charities in general would stop doing this sort of advertising. But I especially wish the most reputable charities would stop. UNICEF and the Red Cross are highly trusted sources for journalists and the public alike, so it's particularly alarming when they publicize claims like 7¢ to "save" a child, or "up to" 3 lives saved per blood donation.
 This stat is common, but extremely dubious. Beyond the obvious "up to" trick, the best case number still appears to be <1. https://medicalsciences.stackexchange.com/questions/1289/how...
I've always taken issue with this type of campaigning, and it makes me want to actively avoid any charity that uses it.
In addition to the author's comment quoted above, it implies there's a long line of children, and they're just waiting for another 7 cent donation so they can vaccinate the next kid.
This is clearly not true, and if it were true, it would make many of the other activities of large charities morally repugnant. How could anyone justify renting offices in major cities, and employing executives on 6-figure salaries, when that money could pay for millions of 7 cent vaccinations?
Obviously, it's a lot more complicated than that. But the fact that the charity is willing to lie to me so easily makes me wonder what else they are lying about.
This seems like silly thinking. I don't think anyone would reasonably believe they put all the money they receive into powder packets. Every $1 you give them will immediately save 14 lives!
They're just saying that's one thing they do with the money, a part of their budget. You should be more concerned with what a given charity's administrative overhead is. How much of that $1 goes towards the $0.07 packets, and how much pays the administrators.
If charity1 has a 20% overhead and then spends the rest on saving the life of a child for every 1000$, while charity2 has a 40% overhead but saves a child for every remaining 500$ spent, which one is better? Overhead is just one factor of a charity's efficiency.
Of course in this case the cost of the powder is just a small factor, and you also need to consider the cost of distribution and other overheads. And the fact that the effectiveness probably drops off at some point. Obviously just spending 1B$ is not going to save 20 billion children, there aren't that many children that need these ORS packets.
If they already have so much money that they buy $0.07 packets for everybody that needs them, then your additional donation won't pay for any additional packets, and you can leave it out of your analysis. Donating won't affect the packets.
But if they don't have enough money to vaccinate everyone, and they're putting as much money into that as they can, then the effect of donating $50 is that three more children are vaccinated.
On the third hand, if they already have enough funding for everything they want to do then your donation is not useful, even if they have low administrative overhead. It won't lead to any extra packets or vaccinations compared to if you didn't donate.
Administrative overhead is good to take into account - if it's very high then something fishy could be going on - but the main thing is the effect of your donation.