Why did Eliyahu tell Elisha to stay?
Malbim: Eliyahu did not want Elisha to be alone with him when Eliyahu departs. Rather, he should be amidst Bnei ha'Nevi'im, so also they will get part of Eliyahu's Ru'ach 1 . Bnei ha'Nevi'im could not be at Eliyahu's departure, for they were not proper to be at that great event 2 . Elisha was not satisfied with this (he wanted to be alone with Eliyahu at the time).
Surely it was not because he did not want Elisha to get a great share of his Ru'ach. Sanhedrin 105b learns from Eliyahu (who later granted Elisha's request for twice Eliyahu's Ru'ach) that one does not envy his Talmid! (PF)
This is unlike Tosefta Sotah 12:5, which says that they were equal to Eliyahu.
