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To start with, most boarding schools, like private and public universities, have an endowment fund, which is essentially their nest egg. The money for the endowment fund comes from alumnae donations, donations from other parties (like the parents of the students, or just general supporters of the school) interest on the fund itself from the bank, and outside investments made from businesses and companies.

Financial aid often has its own funding, but can be tied to the endowment fund. Hogwarts had a financial aid fund:

‘I haven’t got any money,’ [said Tom Riddle]


‘That is easily remedied,’ said Dumbledore, drawing a leather money-pouch from his pocket. ‘There is a fund at Hogwarts for those who require assistance to buy books and robes. You might have to buy some of your spellbooks and so on second-hand, but –’

_{Half-Blood Prince - page 256 - Bloomsbury - chapter 13, The Secret Riddle}

Now it is my understanding that any magical child who is accepted at Hogwarts is able to go there; I don't believe Hogwarts charges tuition, if I recall correctly. The fund for children in need is for books, robes, a wand, etc. If Hogwarts charged tuition, how could five Weasleys possibly attend the school at once (Ginny, Fred, George, Percy, Ron)?

I think, based on canon, that the Ministry does not help fund Hogwarts' expenses, otherwise it would have far too much pull at Hogwarts. The Ministry, aside from Order of the Phoenix, doesn't really interfere at Hogwarts.

There is, however, canon evidence that in the Wizarding world, inventions require patents that must be purchased:

‘Level Seven, Department of Magical Games and Sports, incorporating the British and Irish Quidditch League Headquarters, Official Gobstones Club and Ludicrous Patents Office.’ - The Elevator, Ministry for Magic

_{Order of the Phoenix - page 119 - Bloomsbury - chapter 7, The Ministry of Magic}

I'm not quite sure how patents work in the UK. Here in the US, I believe the inventor who wishes to take out a patent pays for the privilege to do so, from their own private funds. I don't know if any of Dumbledore's inventions were patented. There's the Deluminator, but would he have wanted to disclose the full capabilities of the Deluminator to the patents office? Dumbledore truly doesn't strike me as a man who would even bother to fuss about patents. If he did, though, I'm not sure why anyone other than himself would pay for the patent. As well, any money that came from the patent portion of the sale of an invention would seemingly go to the Ministry as some kind of taxation, with the rest going to the patent holder. I don't think any money would go directly to Hogwarts.

I can't find any evidence in canon that suggests there are taxes in the Wizarding World. I could be entirely wrong on this, though. It kind of goes back to the question on Gringotts on whether or not the Goblins pay interest on their deposits. If there are taxes in the Wizarding World, then undoubtedly some might go to Hogwarts, just as taxes here (US) go to fund education. If it's different in the UK, please leave a comment and I'll adjust this accordingly.

The Malfoys contribute a lot of money to causes that they feel will ingratiate themselves with persons in positions of power, which is mostly the Ministry of Magic. I couldn't find any information on the Malfoys contributing money to Hogwarts, although Lucius Malfoy did provide the Nimbus 2001 brooms for the Slytherin Quidditch team in 1992. Malfoy 1, 2, and 3

There's the Fountain of the Brethren inside the atrium at the Ministry of Magic, which people throw coins into. I would have to figure out how much the average fountain makes in change per year and then convert it from Wizarding currency to pounds/dollars ... But again, there's no guarantee that the money from the fountain would go to Hogwarts.

Hogwarts is home to many magic antiquities, such as the Sword and Gryffindor and the Diadem of Ravenclaw (until it was destroyed in Deathly Hallows), the Sorting Hat, and the jewels in the house hourglasses. I would imagine this is but the tip of the iceberg and that Hogwarts has many, many more priceless items sequestered away. NOTE: This is just my guess, that Hogwarts has a lot of treasure-like items in store. It's not canon.

Canon doesn't specify where Hogwarts' money comes from, as far as I know. So I'm going to suggest that Hogwarts has an endowment fund or the Wizarding equivalent, and that's where its income comes from, to pay teachers, stock the school with supplies, take care of the students, maintain the Hogwarts Express, and provide food and drink for the students.

Remember, also, that they utilize slave labor in the form of house-elves. Can you imagine how much money that must save them? Not a small amount, I'd wager.

In addition to the endowment discussion it seems likely that Hogwarts would also possess substantial land holdings and would generate a significant amount of revenue in rent - given that the school is stated as being around 1000 years old and every witch and wizard of note will have studied there I'm sure it will have received some substantial donations and bequests over the years. This is similarly the case for many ancient boarding schools and universities, for example legend has it that it's possible to walk from Cambridge to London solely on land owned by Trinity College, Cambridge.

In the UK it is completely normal that education is funded by the government, so normal that I would expect J K to assume it. Probably the most definite thing that can be said, given we know nothing about taxes, is "however Wizarding government and bureaucracy is funded, that mechanism also funds Hogwarts". If Hogwarts was entirely funded by some kind of endowment then the Ministry of Magic wouldn't be able to fire the headmaster.

Against that, I know that J K draws heavily on the "boarding school stories" tradition of children's literature - Jennings, Mallory Towers etc. The schools in those stories were invariably fee-paying, though the subject was never mentioned. Those stories, and the assumption that most middle-class kids would go to fee-paying schools, was already outdated when J K was growing up.

On the whole, I think some kind of government funding is most likely.

There is proof that food, at least, is not a problem:

"It's impossible to make good food out of nothing! You can Summon it if you know where it is, you can can transform it, you can increase the quantity if you've already got some - "

_{Deathly Hallows, Chapter 15. Emphasis is mine}

So it is entirely possible that there at least some wizards who don't need to worry about food, and it is extremely likely that Hogwarts would have some of these.

As for pay, what do the teacher's really need other than room and board? Maybe that's how they're paid. As for money for supplies and improving the building & grounds, the students pay for their own stuff, with a few exceptions (Tom Riddle being the only one mentioned), which takes care of the former. As for the latter, there are probably donations, given that nearly all of the wizards and witches in England were taught at Hogwarts.

We know that you can't transfigure food, so you have to grow it or buy it.

There is at least one sentence in the books to disprove this statement:

Mrs. Weasley slammed a large copper saucepan down on the kitchen table and
began to wave her wand around inside it. A creamy sauce poured from the wand tip as she
stirred.

_{Goblet Of Fire, Chapter 5}

We know that you can't transfigure food, so you have to grow it or buy it. Ditto money.

as twoerd said, you can increase the food quantity (magical Banach–Tarski?)

I'm sure that aside from some medicinal plants Herbology Professor had
there were no farmlands at Hogwarts.

I'm not so sure about that; given that not getting noticed is a virtue for house elves they might as well grow food/livestock in not so visible areas (cleared spots in the forbidden forest or Hogwarts fields that are not in the main grounds). Same with Snape's ingredients although some part of those might actually be in his private collection. Given that Hogwarts has been around since the medieval times, it fits that a (basically) castle would get the basic supplies on its own without relying too much on nearby markets; for example, they didn't buy the pumpkins for Halloween, Hadgrid grew them.

Also, we can assume that the plants in the Herbology greenhouse could be sold as potion ingredients after covering Hogwarts' needs.

And they needed to pay the teacher salaries.

While this is true, there is no reason to believe that the teacher salaries are huge (I think that Lupin's clothes were still ragged at the end of the year). After all, all the day-to-day expenses are covered and the position is quite prestigious.

While I'm not saying that Hogwarts has 0 expenses, it doesn't seem that it requires a vast amount of money either.

Just thought I should add: it's unlikely they sell Herbology items and the like. Otherwise, Horace Slughorn would be in deep trouble. He seemed rather sneaky while he was.. "collecting" different items around the school. If the school was funding itself using unicorn hair or the like, Horace would be charged with petty theft at best and full-out embezzlement at the worst. And there won't be any leniency from being Dumbledore's buddy, if the headmaster showed us anything it's that he loves his students. Stealing from his school's funding would NOT go well.

The fact that it is ministry funded is somewhat supported by the events in Book 5. A person is placed at hogwarts to assess the standards which could only be done if it is ministry funded. However, the ministry can not replace the headmaster, near the end of Book 5, it says that the headmasters office was blocked to Umbridge. Additionally, the school has a governing board which means that it is most likely a private school.

In conclusion, hogwarts is a private school, with government funding, with a governing board.