Andrew Kao

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Utility functions without a maximum

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 t to 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.

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.