What is the difference between "to cost" and "to be worth"? Are they mutually replaceable and if not why? For example, is it correct to say that something costs its price.
The cost of an item defines the specific price at which it is sold. It is objective and factual. The sticker price lists the "cost."
The worth of an item is a more intangible sense of value. It may or may not align with the cost. Worth generally aligns with cost when the price is well established and generally agreed upon. However, worth may be subjective, because different people may bring their own opinions into the calculation of worth.
If I have a music box from my grandmother, it would be worth a lot to me (sentimental value) but may not have cost much in the store. Conversely, I recently bought a one-year-old blu-ray player for only a few US dollars. The cost was low, but it was worth much more than that. (This is the scenario where people say things like, "What a bargain!" "It was a steal!")
Yes, it is correct to say that something "costs" its "price." But only you can judge whether something "is worth" its "price."
When worth and cost align, the terms are mutually replaceable. For example, the crude oil market is global, regulated, and well established. I could say "A barrel of oil costs $100" or "a barrel of oil is worth $100" and both would be acceptable.
By contrast, I might see a hand-carved statue that is on sale for $10,000 and say, "It's not worth that much!" It would still cost that much, but I would be making a determination about its worth.
Here's the OED's definition of cost:
That which must be given or surrendered in order to acquire, produce, accomplish, or maintain something; the price paid for a thing.
There are a few subtleties here. First notice that the word money doesn't appear, although that's the thing usually given in a commercial transaction involving the sense of acquire:
A: How much did your new car cost?
B: I paid sticker price, $24,999.
In this sense cost and price are synonymous, and A could well have asked, "What was the price of your new car?"
But in the sense of produce, cost and price are different: cost means the expense of bringing something to sale, and price is what a customer must pay. For going concerns, price must exceed cost. This is reflected in the accounting term of art cost of goods sold, abbreviated COGS, where the cost is the sum of what the producer paid for raw materials and labor.
In investment jargon, the cost of a negotiable instrument is the amount you paid for it when you bought it, and the price is the amount you receive when you sold it. In the US, the tax forms ask you to list the cost of the stock you bought and the price at which you sold it. If the price minus the cost is positive, you will owe taxes on the gain.
In the sense of maintain, cost means expenditures, which may be unrelated to a transaction:
A: What do you want for your birthday?
B: A pony.
A: Do you have any idea what that costs?
The something surrendered may be intangible:
I will beg her to forgive me although the cost will be my pride.
We must keep going whatever the cost.
Finally, there is another term of art from the legal field: costs or court costs are expenditures for counsel and court fees in bringing a law suit.
Worth is the value of something, which may be unrelated to either the cost of that something or its price. The relationship of these words is best illustrated by the following exchange from Oscar Wilde's Lady Windermere's Fan:
Cecil Graham: What is a cynic?
Lord Darlington: A man who knows the price of everything, and the value of nothing.
Cecil Graham: And a sentimentalist, my dear Darlington, is a man who sees an absurd value in everything and doesn’t know the market price of any single thing.