A Jew may not hire workers to do work for him on Shabbat, as a Jew may not ask a non-Jew to do anything for him on Shabbat that he may not do himself. Therefore, a Jew may not hire a non-Jew to work in his field, factory, or store. However, one may hire a non-Jewish worker to help serve food or wash dishes on Shabbat. Since a Jew may do these activities on Shabbat, he may hire a non-Jew to do them. In this case, it is not necessary for the Shabbat payment to be subsumed within a weekday payment (Tehila Le-David 243:1; SSK 28:63; and above 22:14).
If a non-Jew did some work for a Jew on Shabbat, the Jew may not benefit from it on Shabbat. After Shabbat he may benefit from it, but only once enough time has elapsed so that the melakha could have been performed after Shabbat. If the melakha that the non-Jew did for him was done publicly on Shabbat, such as building a home, the Sages decreed that he may never live in the house. However, he may sell it to another Jew. Under pressing circumstances, the first Jew may live in the house the non-Jew built for him (SA 244:3-4 and MB ad loc. 19-20; SA 325:14 and MB ad loc. 73).
All of the above applies to a wage earner. In contrast, a non-Jewish sharecropper or tenant farmer in a Jew’s field may work on Shabbat. Since he shares in the profits, he is working for his own benefit.
Therefore, a Jewish factory owner or store owner may allow a non-Jew to run his business over Shabbat if the non-Jew receives a percentage of the earnings. Even though the Jew is profiting from the work of the non-Jew on Shabbat, since the non-Jew is working to earn money for himself, he is not viewed as working on behalf of the Jew. The Jew may profit from a percentage of this work.
Similarly, a Jew in Israel who owns a field, factory, or store abroad may rent them to a non-Jew in return for a set fee. The non-Jew may then keep the establishment open on Shabbat. Since the Jew gets his rent money in any case, the non-Jew who works on Shabbat is viewed as working for himself. This is on condition that the non-Jew is not only renting on Shabbat, because then it would be clear that the Jew wants the non-Jew to work on Shabbat. Rather, he should rent out the store on a weekly, monthly, or yearly basis, so Shabbat is subsumed within the total.