Which propositions should follow the noun "cost" in different context?

Question

Recently I find myself struggling with choosing the appropriate proposition in a certain context. One of my worst nightmares is the propositions followed by "cost" and how they differ in meanings. The following are specific questions:

1. I know that 'college costs' and 'school costs' mean the money one spends on going to school or going to college, which may involve tuition fees, cost of food and cost of living, etc. However, does "the cost of school" means the money a student or a parent spends on the school or does it mean the money that the school spends on something else to keep it running, such as teachers' and administrators' salaries. Or perhaps to express the latter, we should say " the cost of the school"? Perhaps in the former case, "school/college" represents the act of going to school, while in the latter case, it literally means a school? I feel like my brain is kind of in a knot already. What about "the school's cost"? What does this refer to?

2. I saw these two sentences today in an IELTS writing practise book. They go like this:

Teachers' salaries constitute the largest costs to the school.

The spending on insurance saw a notable upward trend while the costs on books decreased slightly.

(I think there's something off with these two sentences, so I prefer to change the first sentence into: Teachers’ salaries constitute the biggest expenses for the school. and the second into: ...while the costs of books decreased slightly.)

Anyway, that leads me to my second puzzle. I looked up the word "cost" in Longman dictionary (https://www.ldoceonline.com/dictionary/cost), and none of its example sentences match this word with any propositions other than "of", except for in a few rare examples such as "A company hired to do telemarketing ups the cost to as much as 40 percent."

So, does that mean the two sentences in the practice books are wrong? If that is so, what are some more idiomatic ways to express them?

Answer 1

We can't say what the cost of school might mean unless we know whether the sentence is about students or about the school's administration. The former is more likely.

Basically: the cost of x = what you have to pay to buy X

The cost to X = what X has to pay for what is needed.

Costs on is not a common usage; probably the writer used it to avoid using spending twice in the same sentence.

Answer 2

The cost of the school

How expensive the school is to other people.
(Whether this is via tuition fees, or state funding, or both, someone is paying for its running costs)

A cost to the school

How much the school has to pay for something.
(Outgoings - Books, Teacher salaries, etc)

Costs on books

The amount of spending for a particular area (usually an ongoing cost like rent/consumables etc)

raises cost to level

The cost has increased to a specific level
Here, your example actually expands to:

The company hired to do telemarketing has increased the cost (of telemarketing to it's customers) to an amount that is as much as 40 percent higher

The of and to in brackets as the same as above, and the to in bold here is used to indicate direction - where the price has gone.