People are asking the question “How much money do you have to donate to get an expected value of 1 unit of good”.
I think the question is:
How can I do as much good as possible with C units of cost?
This corresponds to the problem of maximising E(U(C)), where U(c) is the utility achieved (via a certain intervention) for the cost c (which must not exceed C). If the budget C is small enough (thinking at the margin):
U(C) = U’(0)*C, where U’(c) is the derivate of U with respect to cost.
Assuming U’(0) and C are independent, mean(“effect”/”cost”) equals mean(“effect”)/mean(“cost”):
mean(“effect”)/mean(“cost”) = E(U(C))/E(C) = E(U’(0))*E(C)/E(C) (assuming independence between U’(0) and C) = E(U’(0)).
So, it seems that, regardless of the metric we choose, we should maximise E(U’(0)), i.e. the expected marginal cost-effectiveness. However, U’(0) and C will not be independent for large C, so I think it is better to maximise mean(“effect”/”cost”).
I think the question is:
How can I do as much good as possible with C units of cost?
This corresponds to the problem of maximising E(U(C)), where U(c) is the utility achieved (via a certain intervention) for the cost c (which must not exceed C). If the budget C is small enough (thinking at the margin):
U(C) = U’(0)*C, where U’(c) is the derivate of U with respect to cost.
Assuming U’(0) and C are independent, mean(“effect”/”cost”) equals mean(“effect”)/mean(“cost”):
mean(“effect”/”cost”) = E(U(C)/C) = E(U’(0)*C/C) = E(U’(0)).
mean(“effect”)/mean(“cost”) = E(U(C))/E(C) = E(U’(0))*E(C)/E(C) (assuming independence between U’(0) and C) = E(U’(0)).
So, it seems that, regardless of the metric we choose, we should maximise E(U’(0)), i.e. the expected marginal cost-effectiveness. However, U’(0) and C will not be independent for large C, so I think it is better to maximise mean(“effect”/”cost”).
Is that a typo on your final bullet point? Should say mean(“effect”)/mean(“cost”).
Hi Stan,
Yes, it was a typo, which I have now corrected. Thanks for catching it! I have also added one extra sentence at the end: