Andrew Kao

Utility functions without a maximum

Yes, agree with you! I was more classifying the *t* that OP was asking about than providing a definition, but wording was somewhat unclear. Have edited the original comment.

Utility functions without a maximum

In economics, if utility is strictly increasing in *t—*the quantity of it consumed—then we would call it a type of a "good", and utility functions are often unbounded. What makes the ultimate choice of *t* finite is that utility from *t* is typically modeled as concave, while costs are convex. I think you might be able to find some literature on convex utility functions, but my impression is that there isn't much to study here:

If utility is strictly increasing in *t* (even accounting for cost of its inputs/opportunity cost etc.), then you will consume all of it that you can access. So if the stock of available *t* is finite, then you have reduced the problem to a simpler subproblem, where you allocate your resources (minus what you spent on *t*) to everything else. If *t* is infinite, then you have attained infinite utility and need not make any other decisions, so the problem is again uninteresting.

Of course, the assumption that utility is strictly increasing in *t, regardless of circumstance,* is a strong one, but I think it is what you're asking.

I see what you mean now, thanks for clarifying.

I'm not personally aware of any "best" or "correct" solutions, and I would be quite surprised if there were one (mathematically, at least, we know there's no single maximizer). But I think concretely speaking, you can restrict the choice set of

tto a compact set of size (0, 1 - \epsilon] and develop the appropriate bounds for the analysis you're interested in. Maybe not the most satisfying answer, but I guess that's Analysis in a nutshell.