What's the different between:
When they're all supposed to mean "to pay money"?
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They don't all mean "to pay money" - in fact none of them mean just that.
"Buy" means "to pay money and obtain something" i.e. you pay money in exchange for the product or service. This is the verb form. In noun form it means the item that was bought.
"Purchase" is a synonym for buy and means the same when used as a verb and noun.
"Buy up" means to buy everything available in the store.
She bought up all the shoes that were on sale.
"Buydown" is a financial term used to mean paying off some part of a loan and reducing interest rates.
A mortgage-financing technique with which the buyer attempts to obtain a lower interest rate for at least the first few years of the mortgage, but possibly its entire life.
Buy-Up - investor's lingo http://www.investopedia.com/terms/b/buy_up.asp
Points paid by a lender to a borrower or mortgage broker for a loan with an above-market interest rate.
but may also refer to buying everything of a given item
Buy-down - mortgage lingo http://en.wikipedia.org/wiki/Buydown
financing technique where the buyer attempts to obtain a lower interest rate for at least the first few years of the mortgage
Purchase Aquire something for money - synonym of Buy
The difference is "buy down" which is a :
a subsidy for a long-term mortgage offered by a third party, as a builder or developer, to lower interest rates for a buyer in the early years of the loan.
Edit: "Buy up" has a nuance that "buy" and "purchase" doesn't have, and that is:
to purchase all, or all that is available, of (something)