If Housecarls are like real-life húskarls, then their livelihood and protection is a duty of their Thane. With that in mind, what is a likely wage that a Housecarl would be making?

In real life, húskarls were sworn by oath to defend their chieftain or king, and were "non-servile manservants or bodyguards". In exchange (in addition to being a fairly honorable position), their chieftain or king was responsible for their wellbeing and the wellbeing of their family, presumably in terms of keeping them fed and clothed and potentially paid.

Monetary questions are always pretty difficult because the game's economy is really not necessarily much like the economy in the lore. However, do we have any indications that Housecarls are paid by the Dragonborn or other Thanes? Does the Jarl arrange to have them paid in your stead, considering they're typically given to the Dragonborn along with a Thane title in exchange for extraordinary services rendered? A Housecarl would likely pull down a more respectable wage than a city guard (and I'd guess that prior to being assigned to the Thane Dragonborn, they WERE city guards that had distinguished themselves in some way).

The game obviously doesn't make you pay for your Housecarls, as it would sort of suck to have 9+ bodyguards all pulling down a wage just to sit in your empty homes. But I'm curious what the obligations of the Dragonborn would be lorewise, if there's any indication.