If I were thinking like an economist, I’d say: You are two rational agents making a bargain. You want someone who wouldn’t require overtime. She thinks that she’s better off relinquishing overtime and keeping this job than pursuing what­ever other options she has available. (I’m bracketing the spend-the-night part for the moment.) So the outcome in which she works for you is better for each of you than the one in which she goes elsewhere and you have to try to find a local person who is as good. What’s to object to?

But I’m not an economist. I’m a moral philosopher. And there’s more going on here than economic arrangements. You and your infant son have a relationship with this woman. It isn’t an arm’s-length transaction, governed by self-interest alone. As it happens, Adam Smith, one of the first great modern economists, was also a great moral philosopher. He wrote “The Wealth of Nations,” in which he explained economic transactions as fundamentally governed by self-interest, and “The Theory of Moral Sentiments,” which explored the role of sympathy in our dealings with others. So let’s enlist both sides of Smith. One question sympathy might lead you to ask is why she’s willing to accept a job with no overtime. The answer, I suspect, is that you and she are operating in an economy in which people doing the kind of domestic work she does often don’t get overtime, even when they’re entitled to it. And you have to decide whether the circumstances of her choice are just.

If the background conditions in a labor market are seriously unjust — as they are in much of the world — people’s willingness to work for a particular wage isn’t enough to make that wage fair. We exploit people when we take advantage of their vulnerabilities. That’s why many think that American companies can’t defend the wages they pay in Bangladesh by saying that the workers there are willing to accept them. You’re concerned that you’d be taking advantage of her by accepting the terms she’s offering, and there’s good reason for that.

Adam Smith lectured on legal theory as well. So he would have pointed out that you also have legal constraints here. Because people in her situation tend to be in a weak bargaining position, federal and a few state statutes create widely applicable overtime provisions. But the costs of commuting are generally covered by the employee, and the time spent doing it is not paid time. From that point of view, if she stays overnight on Friday and is not working, you are, in effect, allowing her to avoid the cost and the inconvenience of a long commute late Friday, followed by an early commute back, if she’s to work on Saturday morning. Giving up the discretionary payment of $50 would seem reasonable on her part, in view of the elements of convenience to her.

There’s one more factor that’s ethically relevant. Simply ignoring her own judgments about what options are best for her would be less than respectful. It seems to me that you can discuss all this with her and come to a judgment together about what terms you both think are fair. That might still be somewhat more than what you would pay to a new caretaker, because you recognize, out of sympathy, the burden of her long commute; and because there’s a final argument of self-interest here: You know this woman and you trust her alone in your house with your son. That peace of mind has a value, too.