One of the issues in Bryan Caplan’s famous “Why I’m Not An Austrian Economist” essay (even though he had been one when he was younger) is the issue of cardinal utility functions. A lot of Rothbardians like to roll their eyes at the mainstream for thinking utility is a cardinal entity that can be measured in principle, whereas Austrians know that preferences are ordinal rankings. In a fit of exasperation, the Rothbardian might exclaim that the standard equilibrium conditions in a typical micro book involve dividing by marginal utility!!
But, as Bryan points out in his essay, such jabs misconstrue what the mainstream economists are doing. They don’t actually take a particular utility function seriously; it merely “represents” ordinal rankings over bundles of goods. There’s no special significance if someone has a certain utility function, because you could represent the same ordinal preferences with any monotonic transformation of that same function. (NB: If you’re doing von Neumann Morgenstern expected utility functions then it can only be a positive affine transformation.) Bryan is right, by the way: When I was a young punk down in Auburn my first year, I made sure everybody knew that I knew a bunch of math from NYU.
Anyway, somewhat apropos, today at EconLog Bryan says that there’s little doubt that a<1, when we model spouses' utility function as U=(Family Income)/(2^a). But Bryan wanted to know: "Where does a typically lie in the real world?” Daniel Kuehn reminded in the comments:
“Where does ‘a’ typically fall in the real world” isn’t a particularly meaningful question because it’s a monotonic transformation. Again I think the private/household good entering a utility function is the better approach.
So then the question is maybe something like what is the marginal rate of substitution between some representative (or bundle of… but I’d have to think about MRS’s of bundles) rivalrous and non-rivalrous goods.