Would it be correct to say that any lost item whose value is lower than the value of the time needed to deal with returning it (based on the average wage in the economy) does not have to be returned? That is, suppose a person sees a lost item worth X. And suppose the value of the time needed to handle it is 50 shekels, the average hourly wage in the economy, times an hour and a half of dealing with returning it, for a total handling cost of 75 shekels. If the value of the lost item, X, is less than 75 shekels, is there no obligation to return it?

Obviously. The Talmud explicitly says that in such a case a person sells the lost item and keeps the money for the owner. He is exempt from dealing with it. It is the same reasoning of “there shall not be a needy person among you,” but really one need not even get that far. It is a simple logical consideration: my fellow’s lost property is not preferable to my own lost property. Your life takes precedence over your fellow’s life.

Moreover, the owner has to pay my expenses, and if he would have to pay me more than the value of his lost item, he himself would not want that.

The question is whether the time spent dealing with the lost item (putting up signs, etc.) can be considered time for which the person handling it deserves payment, even though he can do it outside his working hours so that he is not losing wages.

To illustrate, suppose I see a scarf someone lost that is worth 20 shekels. Putting up the signs would take me, say, an hour. The question is whether my hour is more important than the other person’s 20 shekels (assuming I would not suffer any monetary wage loss as a result of handling it, but only lose an hour in which I could have studied), or perhaps we should say that since this is an hour in which I am occupied with a commandment, there is not really any loss of an hour here.

If he is not losing wages, then he is not entitled to payment. You did not spend money on him, only labor, and the commandment is worth that labor. It is your obligation. Prevented profit counts as a loss, and therefore you are entitled to compensation for that as well. But if no profit was prevented from you, there is no obligation to pay you.

From the fact that I could have worked during that time and chose not to, I have effectively revealed that this time is worth x money to me. Therefore losing that time is, from my perspective, equivalent to losing at least x money. If so, why should I lose that time in order to save someone else less than x money?

In other words, loss of time is equivalent to monetary loss.

Because in order to perform commandments we are required to invest time. In returning lost property we are exempt from investing money if the investment is greater than the other person’s gain, because there is also returning lost property to myself, and he is not preferable to me (and he would also have to pay me for my loss, so it is not worthwhile for him either). But investing time for the commandment of returning lost property is no less obligatory than investing the time required to build a sukkah. What is the difference between this and that?

Although just now I thought there may be room for a more complex argument, not because of the value of time but because of comparing three alternatives. Suppose I can earn 100 shekels an hour. Now I chose not to work in order to write a book. And now I encounter the commandment of returning lost property, and I say to myself that if I had known in the end that I would not write the book, I would have gone to work. So why can returning lost property force me not to write the book? Seemingly I am not obligated to return it, since I could have gone to work and thereby exempted myself from the return.
It seems to me there is room for the reasoning that if in fact I really would have gone to work had I not been writing the book—then I am indeed exempt from returning it in such a case. But if I am not going to work simply because I do not feel like it, and I am using the opportunity to write a book, then the fact that I could have earned 100 shekels is not considered a loss for which one must pay me and which exempts me from returning it.

This reminds me a bit of the paradox of matzah made from new grain. A person is obligated to eat matzah on Passover (a positive commandment). But the flour is very expensive; suppose flour for matzah costs half his wealth, and a person need not spend more than a fifth of his assets on a positive commandment. So he is exempt. Now there is an option to buy flour from the new grain (that year’s grain, which is forbidden before the day of waving, that is, the first day of the intermediate days of the festival), and it is cheap. So perhaps I should buy it, since the positive commandment of matzah overrides the prohibition of new grain. But in order not to violate a prohibition I must spend all my wealth, so I will buy from the old grain instead (for half my wealth). But then it turns out I spent half my wealth in order to fulfill a positive commandment, since I could have bought nothing and not eaten matzah.

You wrote that returning lost property is similar to the commandment of sukkah regarding investment of time. The question is whether there is room to distinguish between commandments that belong to the area of Choshen Mishpat and commandments from Orach Chayim / Yoreh De’ah. The distinction I am proposing is that in Choshen Mishpat commandments there is more room to invoke the reason for the verse, and where the reason no longer applies, the commandment no longer applies either. If we say that the reason for the commandment of returning lost property is that I must treat my fellow’s money as I treat my own money, and if we add the assumption that my own money takes precedence over my fellow’s money, then it follows that the commandment of returning lost property exists only where the loss (in time or money) required of me in order to save my fellow’s property is less than the value of his lost item. For example, if a smartphone worth 3,000 shekels is lost, then certainly I must make an effort to return the lost item to my fellow, but if something of relatively lower value was lost, worth less than 50 shekels, then there is room to say that there is no obligation to return his lost item, because the reason no longer applies.

That analysis does not seem right to me. Even if your fellow’s money is as important to you as your own, you are still allowed to forgo your own money, but not your fellow’s money.

I looked into this question again (especially your response beginning with the words “Because in order to perform commandments we are required to invest time”), and I do not understand why time (and also information and honor) cannot be treated as resources to which returning lost property applies. After all, the Sages also include restoring a person’s body from the verse about returning lost property. If so, one could conclude that returning lost property also applies to things that are not actual money (such as time, information, and honor). Likewise, in Bava Metzia 31a it says:
Rava said: “Every lost thing of your brother” — this comes to include loss of land. Perhaps one could also include here loss of time, information, and honor (and in general anything that can be lost).
In addition, time is something worth money, because one can estimate how much a person would be willing to pay in order to avoid investing X amount of time in the business of returning lost property, and thus assess the value of that time. Payments for pain and suffering are assessed similarly. This is what Rashi brings in Bava Kamma 4:
Pain — we assess how much a person whose hand is about to be cut off by a sword would want to pay in order for it to be cut off by a drug, where there is not so much pain.
And in general there is the well-known rule that time is money. The same is true regarding information and honor: one can estimate how much a person would be willing to pay to recover information/honor lost to him (for example, like a password to a Bitcoin wallet that he forgot, or evidence unknown to him that would exonerate a defendant in a trial involving disgrace). Perhaps that is also the reason for the law of an elderly person for whom it is beneath his dignity being exempt from returning lost property (for his own comes before anyone else’s).
In the Mishnah in tractate Bava Metzia, chapter 2, it says:
“If he was idle from work worth a sela, he may not say to him: Give me a sela; rather, he gives him his wage as an idle worker. If there is a court there, he makes a stipulation before the court; if there is no court there, before whom can he stipulate? His own takes precedence.”
Rabbi Ovadiah of Bartenura explains:
“As an idle worker” — how much a person would want to take, and have deducted from his wage, in order to refrain from this heavy labor in which he is engaged and instead do such light work.
“If there is a court there” — if he does not wish to refrain from his work, whose pay is high, what should he do? If there are three people there, he stipulates before them and says: See that I earn such-and-such, and I do not wish to stop and take only a small wage; if you will say that I may take my full wage, I will deal with returning this lost item.
“His own takes precedence” — and he leaves the lost item.

And in Maimonides, Laws of Robbery and Lost Property 12:4:
Likewise, if he was engaged in work and ceased from his work worth a dinar, and he returned a lost item worth a hundred dinars, he may not say to him, “Give me the dinar that I lost”; rather he gives him his wage as an idle worker, who would cease from that work in which he was engaged. But if he stipulated with the owners, or before a court, that he would take what he loses, and they authorized him, then he takes it. And if there is no court and no owners, his own takes precedence.

I think it follows from here that one should assess how much a person wants to receive in order to stop his activities (even ordinary activities like reading a book or walking in the park) and engage in the work of returning lost property, and if that amount is greater than the wage of an idle worker, then if he stipulated with the owners or before a court that he would take what he would lose (according to the estimate above) and they authorized him, then he takes it; and if there is no court and no owners, his own takes precedence, meaning that in such a case it is relevant to say that his own lost property takes precedence over his fellow’s, and he leaves the lost item.

I am not sure about that comparison. After all, you gain a commandment and its reward. Time is the resource you are required to invest, but you get something in return, and perhaps that too has to enter the calculation.
Beyond that, I am not sure you can claim after the fact that your time is worth a great deal (beyond that of an ordinary person), unless it is known that this is indeed always the case for you. Moreover, stipulating before a court helps only for receiving the value of the time you actually lost (that is, how much you would in fact have earned). We do not find stipulating before a court in order to receive a lot of money for idleness because in your opinion your time is worth a lot of money. Therefore stipulating before a court would not help here either. On the contrary, the very fact that they brought the stipulation only regarding someone who was actually engaged in work proves against your reasoning, that even a stipulation would not help in such a case, and all the more so without a stipulation.

Regarding the reward for the commandment: if one takes it into account with respect to the resource of time, one should also take it into account with respect to ordinary wages.
Each person assesses the value of his time differently, just as each person assesses the value of his pain differently, and just as each person earns differently. But even if we say not, one can make a calculation of the value of time with respect to the ordinary person—namely, how much the ordinary person would want to receive in order to stop his activities and engage in the work of returning lost property.
I think the court serves as the lost owner’s representative and represents his interest (like a board of directors of a public company that represents the interest of the investing public even though it was not appointed directly by them and not even by election). Just as I can stipulate with the owner regarding the value of my idle time, so too I can stipulate with a court.
Regarding your last point, that they brought the stipulation only regarding someone who was actually engaged in work, I think that does not necessarily mean only someone engaged in work, but anyone who loses or is prevented from profit if he has to deal with returning the lost item. Being engaged in work is only a common example of prevented profit, but even someone who was in the middle of a trip with his children, for example, and had to interrupt the outing in order to return a lost item, has prevented profit here too (the “profit” being the enjoyment he gets from family time). Therefore one should assess how much he is willing to take in order to interrupt this enjoyment, and that is what he is entitled to (provided that he stipulated this with the owners or the court).

I think the whole problem begins from the fact that in the reality of the Sages, the equation “time = money” was less relevant, because they had much more free time than we do, and on the other hand far fewer possibilities of paid employment. Nowadays the average person works for most of his waking hours. In other words, free time has become a rarer and therefore more expensive resource, and at a certain point in history (perhaps beginning with the Industrial Revolution) it began to acquire value like any other resource worth money. From the fact that a person chose to work, say, fewer overtime hours in order to have more free time, he has revealed his view that his free time is preferable to wages. Therefore, if he encounters a lost item during his free time, that should be no worse than encountering a lost item during work hours (similar to the example you gave above about someone who gave up a job paying 100 shekels an hour in order to write a book).

There is room for that reasoning, but the matter still requires further thought.