Following up on this, i did work out a solution for the case where we have an annual income, there is outside funding, there’s no discounting, and there’s a consumption option where instead of funding a project we can just collect K utils per dollar. The work behind this is a mess, partly because the equations get long and partly because it was mostly just me doing the same thing repeatedly for slightly different situations until i stopped being confused. Since either declining to show my work or putting 14 kilobytes of garbage in a forum comment would both be bad, here it is in a pastebin link: https://pastebin.com/raw/PnDZ2rTZ
The result is if there are two projects, X and Y, and our income I is such that we can’t affect which of two projects gets done first, that is, if C_X / F_X < C_Y / (F_Y + I), then project X will always be finished before project Y and there’s nothing we can do about it, then we should fund whichever project has higher U/F. But if we are able to affect which project gets done first, we should fund whichever has higher (U—K*F) / C.
And after thinking about it more and writing more equations, i think U/F really does give us a direct comparison of project-like interventions (utility over time forever once it’s fully funded) to consumption-like interventions (utility per dollar). And it gives us a direct comparison of project-like interventions if we can spend money to complete a project in zero time. And it gives us a direct comparison of project-like interventions in the case where making/spending money takes time, but that time doesn’t matter because we can’t change the order in which things get done. The case where it does not work is when we have an annual income as opposed to a one-shot budget and we’re comparing two project-like interventions and the one that we fund first is the one that gets done first.
I think what makes the result for the case where we have an income and can determine which intervention gets done first so qualitatively different from the case where have a stack of cash and can choose between two projects to knock out is that we have to take into account how much choosing to fund the first project delays our ability to fund the second project. And that delay is proportional to C. (Everything here is assuming no diminishing returns to rate of funding, so it’s always best to concentrate funding on one project to knock it out as soon as possible and never makes sense to split funding between projects and get neither one done.)
Following up on this, i did work out a solution for the case where we have an annual income, there is outside funding, there’s no discounting, and there’s a consumption option where instead of funding a project we can just collect K utils per dollar. The work behind this is a mess, partly because the equations get long and partly because it was mostly just me doing the same thing repeatedly for slightly different situations until i stopped being confused. Since either declining to show my work or putting 14 kilobytes of garbage in a forum comment would both be bad, here it is in a pastebin link: https://pastebin.com/raw/PnDZ2rTZ
The result is if there are two projects, X and Y, and our income I is such that we can’t affect which of two projects gets done first, that is, if C_X / F_X < C_Y / (F_Y + I), then project X will always be finished before project Y and there’s nothing we can do about it, then we should fund whichever project has higher U/F. But if we are able to affect which project gets done first, we should fund whichever has higher (U—K*F) / C.
And after thinking about it more and writing more equations, i think U/F really does give us a direct comparison of project-like interventions (utility over time forever once it’s fully funded) to consumption-like interventions (utility per dollar). And it gives us a direct comparison of project-like interventions if we can spend money to complete a project in zero time. And it gives us a direct comparison of project-like interventions in the case where making/spending money takes time, but that time doesn’t matter because we can’t change the order in which things get done. The case where it does not work is when we have an annual income as opposed to a one-shot budget and we’re comparing two project-like interventions and the one that we fund first is the one that gets done first.
I think what makes the result for the case where we have an income and can determine which intervention gets done first so qualitatively different from the case where have a stack of cash and can choose between two projects to knock out is that we have to take into account how much choosing to fund the first project delays our ability to fund the second project. And that delay is proportional to C. (Everything here is assuming no diminishing returns to rate of funding, so it’s always best to concentrate funding on one project to knock it out as soon as possible and never makes sense to split funding between projects and get neither one done.)