daf bit: Ketubot 108

The mishna records the following disagreement. If a man went to a country beyond the sea, leaving his wife behind, and another man came forward and maintained her in her husband's absence, does that man have a claim upon the husband? Chanan says no, he loses his money, while the sons of the high priests (I don't know who they are) say he can recover it if he takes an oath concerning the amount he spent. R. Dosa b. Harkinas (a name I don't know) agrees with them, while R. Yochanan b. Zakkai says: Chanan spoke well; the man has placed his money on a stag's horn and it is unrecoverable.

The g'mara analyzes the argument. Elsewhere we learn that a man can pay someone else's shekel (Temple tax), repay his debt, or restore to him an object he has lost, all without benefit to himself, meaning he is not repaid. The shekel and restoring lost objects are religious duties, and repaying debt (some say, but this is disputed) spares the man shame. The g'mara does not here close the loop by addressing whether maintaining the man's wife in his absence also spares him from shame. (107b-108)

The talmud here does not raise the question of agency; if the husband had asked somebody to take care of his wife while he was away, that would be a different case.