Who were the poor wise youth and the old foolish king?

The chapter doesn't say. It names neither of them, and it doesn't say when or where. What it gives is the saying and what followed: "Better is a poor and wise youth than an old and foolish king who doesn’t know how to receive admonition any more" (4:13); "For out of prison he came out to be king" (4:14); "I saw all the living who walk under the sun, that they were with the youth, the other, who succeeded him" (4:15). And the Preacher's word on it: "yet those who come after shall not rejoice in him. Surely this also is vanity and a chasing after wind" (4:16). The chapter doesn't say whether the youth of 4:15, "the other", is the youth of 4:13, or whom he means in 4:14 and him in 4:16, and these notes don't decide it. They don't name the youth or the king, and don't make either of them a picture of anyone.