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.
Infinite t does not necessarily deliver infinite utility.
Perhaps it would be simpler if I instead let t be in (0, 1], and U(t) = {t if t < 1; 0 if t = 1}.
It’s the same problem, with 1 replacing infinity. I have edited the question with this example instead.
(It’s not a particularly weird utility function—consider, e.g. if the agent needs to expend a resource such that the utility from expending the resource at time t is some fast-growing function f(t). But never expending the resource gives zero utility. In any case, an adverserial agent can always create this situation.)
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.
In economics, we call something a “good” if utility is increasing in t—the quantity of it consumed—and utility functions are often unbounded
I don’t think that’s true. Modern economists say utility is increasing in t _at the relevant margins_. There is no claim that it’s unbounded or that it doesn’t become negative utility at some future time or quantity.
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.
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.
Infinite t does not necessarily deliver infinite utility.
Perhaps it would be simpler if I instead let t be in (0, 1], and U(t) = {t if t < 1; 0 if t = 1}.
It’s the same problem, with 1 replacing infinity. I have edited the question with this example instead.
(It’s not a particularly weird utility function—consider, e.g. if the agent needs to expend a resource such that the utility from expending the resource at time t is some fast-growing function f(t). But never expending the resource gives zero utility. In any case, an adverserial agent can always create this situation.)
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.
I don’t think that’s true. Modern economists say utility is increasing in t _at the relevant margins_. There is no claim that it’s unbounded or that it doesn’t become negative utility at some future time or quantity.
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.