1. When you claim that X is most efficient
way for the federal government to spend an extra billion dollars, you should point to a large externality you propose to fix - and argue that the supply response will be substantial. Answers of the form "Distribute the money equally to everyone" are weak, because all lump-sum transfers are neutral with respect to Kaldor-Hicks efficiency.
2. When you claim that X is the maximally utilitarian way for the federal government to spend an extra billion dollars, you should start with Kaldor-Hicks efficiency, then adjust for marginal utility of wealth. The conventional adjustment is to put higher weight on the poor, but there are equally strong cases for putting higher weight on the appreciative (people who savor better experiences) and the materialistic (people who place unusually high value on the stuff that money can buy). The true utilitarian will want to make each and every one of these adjustments.
I agree with RPLong on a simple shape-of-utility curves
basis that redistribution is the obvious first-cut winner (and do it
progressively). Anything else needs to overcome normal consumer theory
with an interesting externality or multiplier story.
I think investments in neglected R&D of any sort would be a strong contender. A billion goes a long way in a prize fund too.
Endow a fund that spends 5% of the endowment per year on prizes for solving specific scientific problems.
A very good start, but you should specify "specific scientific problems" with especially large externalities (for maximum efficiency) and especially large effects on the poor, appreciative, and materialistic (for maximum utility).
There are several enticing proposals to spend the money improving public opinion on e.g. immigration. But how much difference would $1B of open borders propaganda really make on policy?
Sending medicine and vaccinations to the third world is probably the way
to provide the greatest happiness to the greatest number for the lowest
cost. The domestic welfare state by and large isn't because people in
the US tend to have a lot of money by world standards anyway, and
development aid (as opposed to relief of extreme hardship) isn't either
because it doesn't seem to do much of anything.
Last but not least, Floccina gets credit for citing Alex Tabarrok.