Does my saying under oath, "The shoe's capital is more than I paid,"—knowing full well that it is exactly the same—count as a false oath (yameen kadhiba) that necessitates expiation (kaffara)? And what is the expiation if it is due?

Informing someone that the price of a shoe on display is less than its cost, while in reality it is not, is considered a lie for which repentance is obligatory. As for swearing falsely on purpose, it is a great sin, and the majority of scholars hold that there is no expiation for it, but rather repentance is obligatory. Imam Shafi'i, however, believes that expiation is obligatory, and combining repentance with expiation is more cautious. If the oath was unintentional, it falls under the category of a vain oath (laghw al-yamin) for which there is no expiation. Swearing an oath in a sale is disliked if truthful, and a major sin if false.