Vasco Grilo🔸 comments on Probability distributions of Cost-Effectiveness can be misleading

[Vasco Grilo🔸]

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”/​”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”).

[Stan Pinsent]

Is that a typo on your final bullet point? Should say mean(“effect”)/​mean(“cost”).

[Vasco Grilo🔸]

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:

However, U’(0) and C will not be independent for large C, so I think it is better to maximise mean(“effect”/​”cost”).