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The castle is under siege. The inhabitants are running out of food but have plenty of gold.

The attackers have food supplies but need money to keep them coming.

Does it make sense for the attackers to sell food to the castle-dwellers at inflated prices in order to get all their gold rather than simply starving them out at the beginning?

If this is so then why has no-one ever done it?

I think perhaps I should say something about why I think it's a good idea.

If you simply starve the castle-dwellers from the start then they will likely bury all their valuables in the hope of recovering them later. They will also start eating valuable animals such as horses.

If you give them food in exchange for their valuables of all kinds in return for food then you have both the gold and the valuables. Once they no longer have anything worthwhile to pay you with, then you can start to starve them out.

It makes no sense, as the attackers will obtain all the gold of the besieged anyway, as soon as the defense has fallen. Selling food to the defenders is the worst possible action, as it increases the losses of the attackers who need to resupply, keep their troops healthy and motivated without being able to do much. Time is essential in a Siege. The longer it takes, the higher the risk of winter, reinforcements, changing alignments breaking the siege. And hunger is your best weapon, breaking your enemies' resolve, morale and ability to fight most reliably and within a reasonable amount of time.

This would be a good thing for a story.

In general, those in charge who are directing a siege want to starve out the occupants of the city under siege. This is because the besieged site for whatever reason cannot be taken by direct military action.

The troops who are actually conducting the siege might have very different motivations. These commoners might have been drafted to fight, compelled to stay by force or stay in the hope of getting loot when the city falls. Individuals among these common soldiers might very well surreptitiously take the chance of selling food to persons in the city under siege. These individuals make some profit in the short term - which makes sense if maybe the city is not going to fall or for some reason they are not going to get any loot. If someone catches one of these opportunists, that person might be in trouble - or might just have to hand over half the gold...

Blockade runners could be considered an example of this. A naval blockade is a type of siege. A port city dependent on resupply by sea is prevented from resupply by the blockading naval force. The blockade runners evade the blockade to resupply to city. Could the blockade runners be members of the same navy that is doing the blockade? A wily captain might try to butter his bread on both sides this way.

This only makes sense if the gold is what the attackers want

If the attackers are looking for land or political influence, then this doesn't make sense at all. Why prolong the engagement if the gold isn't the #1 reason you're there? Why risk your people and your own supplies?

If the gold is what you want, selling food until you have all the gold is a dandy solution. If the gold isn't the single most important reason you're there, this is a remarkably foolish tactic.

In tribal warfare trading with the enemy is normal, even in the modern age in conflicts like the ongoing Syrian war. The factions were trading fuel and weapons, even though those same weapons would likely be turned against them. So the concept itself is not entirely irrational. It's very much a matter of the sort of war you're having.

However, half the purpose of a siege is to starve the castle out, so you're not going to be trading them food. Once the castle runs out of food you've won, so you're not going to act in a way that extends the siege beyond the minimum necessary, regardless of any other potential returns.

If you're worried about them burying the gold, simply force them to ransom themselves. They have to give you the gold in exchange for free passage out.

This is Normal

First of all, this is the case in so many other contexts as well. "Your money or your life" If the armed robber is really ready to kill someone, they won't even ask the question. They do ask the question because they don't really want to kill someone, they just want the money. But they are ready to kill if that's the only way to get the money.

Similarly, the attackers don't really want to kill people, or have them die of starvation. They'd rather get the money, the castle and slaves. So they are perfectly content to get the gold and wait on the castle. The people willing to trade rather than starve are the same people who will, in the end, rather surrender and become slaves than die defending the castle. Well, the leaders will probably get killed anyway, as leaders tend not to make great slaves, but most of the people in the castle were effectively slaves to the owners of the castle anyway, so they'll gladly become slaves of the other side instead of dying of starvation if that is an option.

Where do the attackers get the food? From the peasants surrounding the castle who have already been beaten into submission. So they don't actually spend the gold on food, they spend it on luxuries and send part of it (reportedly "all") home to their king.

For those who say that they would never do such a thing, keep in mind that a common way of managing far-flung lands was to have them pay tribute (a.k.a. exorbitant taxes), so getting a steady income stream from the castle was really just a prelude to the long-term situation. (Until the revolt...)